Hrithik Roshan is her family. Hrithik Roshan: personal life

Some definitions:

1. Polyhedron represents geometric body, limited to a finite number of planar polygons, any two of which having a common side do not lie in the same plane. In this case, the polygons themselves are called faces, their sides are called edges of the polyhedron, and their vertices are called vertices of the polyhedron.
2. The figure formed by all the faces of a polyhedron is called its surface ( full surface), and the sum of the areas of all its faces is (total) surface area.
3. is a polyhedron with six faces, which are equal squares. The sides of the squares are called the edges of the cube, and the vertices are called the vertices of the cube.
4. is a polyhedron with six faces and each of them is a parallelogram. The sides of parallelograms are called the edges of the parallelepiped, and their vertices are called the vertices of the parallelepiped. The two faces of a parallelepiped are called opposite, if they do not have a common edge, and those having a common edge are called adjacent. Sometimes some two opposite faces of a parallelepiped are highlighted and called reasons, then the remaining faces are side faces, and their sides connecting the vertices of the bases of the parallelepiped are its lateral ribs.
5. Right parallelepiped- this is a parallelepiped whose side faces are rectangles. is a parallelepiped whose all faces are rectangles. Note that every rectangular parallelepiped is a right parallelepiped, but not every right parallelepiped is a rectangular one.
6. opposite. The segment connecting the opposite vertices of a parallelepiped is called diagonally parallelepiped. A parallelepiped has only four diagonals.
7. Prism ( n-coal) is a polyhedron with two equal faces n-gons, and the rest n faces are parallelograms. Equals n-gons are called reasons, and parallelograms – lateral faces of the prism- This is a prism whose side faces are rectangles. Correct n-carbon prism- this is a prism in which all the side faces are rectangles, and its bases are regular n-gons.
8. The sum of the areas of the lateral faces of the prism is called its lateral surface area(denoted S side). The sum of the areas of all faces of a prism is called prism surface area(denoted S full).
9. Pyramid ( n-coal)- this is a polyhedron, which has one face - some n-gon, and the rest n faces – triangles with a common vertex; n-a square is called basis; triangles that have a common vertex are called side faces, and their common vertex is called the top of the pyramid. The sides of the faces of a pyramid are called its ribs, and the edges converging at a vertex are called lateral.
10. The sum of the areas of the lateral faces of the pyramid is called lateral surface area of ​​the pyramid(denoted S side). The sum of the areas of all the faces of the pyramid is called surface area of ​​the pyramid(surface area is denoted S full).
11. Correctn-coal pyramid- this is a pyramid whose base is correct n-gon, and all side edges are equal to each other. A regular pyramid has side faces that are equal to each other isosceles triangles.
12. The triangular pyramid is called tetrahedron, if all its faces are equal regular triangles. A tetrahedron is a special case of a regular triangular pyramid (i.e., not every regular triangular pyramid will be a tetrahedron).

Axioms of stereometry:

1. Through any three points that do not lie on the same line, there is a single plane.
2. If two points of a line lie in a plane, then all points of the line lie in this plane.
3. If two planes have a common point, then they have a common straight line on which all the common points of these planes lie.

Corollaries from the axioms of stereometry:

• Theorem 1. A single plane passes through a straight line and a point not lying on it.
• Theorem 2. A single plane passes through two intersecting lines.
• Theorem 3. A single plane passes through two parallel lines.

Construction of sections in stereometry

To solve problems in stereometry, it is urgently necessary to be able to construct cross-sections of polyhedra (for example, a pyramid, parallelepiped, cube, prism) in a drawing using a certain plane. Let's give a few definitions to explain what a section is:

• Cutting plane pyramid (prism, parallelepiped, cube) is such a plane, on both sides of which there are points of a given pyramid (prism, parallelepiped, cube).
• Cross section of a pyramid(prism, parallelepiped, cube) is a figure consisting of all points that are common to the pyramid (prism, parallelepiped, cube) and the cutting plane.
• The cutting plane intersects the faces of the pyramid (parallelepiped, prism, cube) along segments, therefore section there is a polygon lying in the cutting plane, the sides of which are the indicated segments.

To construct a section of a pyramid (prism, parallelepiped, cube), you can and should construct the intersection points of the cutting plane with the edges of the pyramid (prism, parallelepiped, cube) and connect every two of them lying on the same face. Note that the sequence of constructing the vertices and sides of the section is not significant. The construction of sections of polyhedra is based on two construction tasks:

1. Lines of intersection of two planes.

To construct a straight line along which some two planes intersect α And β (for example, a cutting plane and a polyhedron face plane), you need to construct two of their common points, then the straight line passing through these points is the line of intersection of the planes α And β .

1. Points of intersection of a straight line and a plane.

To construct the intersection point of a line l and planes α need to construct the point of intersection of the line l and straight l 1 along which the plane intersects α and any plane containing a line l.

The relative position of straight lines and planes in stereometry

Definition: When solving problems in stereometry, two straight lines in space are called parallel, if they lie in the same plane and do not intersect. If straight A And b, or AB And CD are parallel, then they write:

A few theorems:

• Theorem 1. Through any point in space not lying on a given line, there passes a single straight line parallel to the given line.
• Theorem 2. If one of two parallel lines intersects a given plane, then the other line also intersects this plane.
• Theorem 3(sign of parallel lines). If two lines are parallel to a third line, then they are parallel to each other.
• Theorem 4(about the point of intersection of the diagonals of a parallelepiped). The diagonals of a parallelepiped intersect at one point and are bisected by this point.

There are three possible cases relative position straight line and plane in stereometry:

• The straight line lies in the plane (every point of the straight line lies in the plane).
• A straight line and a plane intersect (have a single common point).
• A straight line and a plane do not have a single common point.

Definition: A straight line and a plane are called parallel, if they do not have common points. If straight A parallel to the plane β , then they write:

Theorems:

• Theorem 1(a sign of parallelism between a line and a plane). If a line not lying in a given plane is parallel to some line lying in this plane, then it is parallel to the given plane.
• Theorem 2. If the plane (in the figure – α ) passes through a straight line (in the figure – With), parallel to another plane (in the figure – β ), and intersects this plane, then the line of intersection of the planes (in the figure - d) is parallel to this line:

If two different lines lie in the same plane, then they either intersect or are parallel. However, in space (i.e. in stereometry), a third case is also possible, when there is no plane in which two straight lines lie (and they neither intersect nor are parallel).

Definition: Two straight lines are called interbreeding, if there is no plane in which they both lie.

Theorems:

• Theorem 1(sign of crossing lines). If one of two lines lies in a certain plane, and the other line intersects this plane at a point not belonging to the first line, then these lines intersect.
• Theorem 2. Through each of two intersecting lines there passes a single plane parallel to the other line.

Now let's introduce the concept of an angle between skew lines. Let a And b O in space and draw straight lines through it a 1 and b 1, parallel to straight lines a And b respectively. Angle between intersecting lines a And b called the angle between the constructed intersecting lines a 1 and b 1 .

However, in practice the point O more often they choose so that it belongs to one of the lines. This is usually not only more convenient, but also more rational and correct from the point of view of constructing a drawing and solving a problem. Therefore, for the angle between crossing lines we give the following definition:

Definition: Let a And b- two intersecting straight lines. Let's take an arbitrary point O on one of them (in our case, on the straight line b) and draw through it a straight line parallel to the other of them (in our case a 1 parallel a). Angle between intersecting lines a And b is the angle between the constructed line and the line containing the point O(in our case this is the angle β between straight lines a 1 and b).

Definition: Two straight lines are called mutually perpendicular(perpendicular) if the angle between them is 90°. Both straight lines that intersect and straight lines lying and intersecting in the same plane can be perpendicular. If straight a perpendicular to a straight line b, then they write:

Definition: The two planes are called parallel, if they do not intersect, i.e. have no common points. If two planes α And β are parallel, then, as usual, they write:

Theorems:

• Theorem 1(a sign of parallel planes). If two intersecting lines of one plane are respectively parallel to two lines of another plane, then these planes are parallel.
• Theorem 2(about the property of opposite faces of a parallelepiped). The opposite faces of a parallelepiped lie in parallel planes.
• Theorem 3(about the straight lines of intersection of two parallel planes with a third plane). If two parallel planes are intersected by a third, then their intersection lines are parallel to each other.
• Theorem 4. Segments of parallel lines located between parallel planes are equal.
• Theorem 5(about the existence of a unique plane parallel to a given plane and passing through a point outside it). Through a point not lying in a given plane there passes a single plane parallel to the given one.

Definition: A line intersecting a plane is called perpendicular to the plane if it is perpendicular to every line lying in this plane. If straight a perpendicular to the plane β , then they write, as usual:

Theorems:

• Theorem 1. If one of two parallel lines is perpendicular to the third line, then the other line is also perpendicular to this line.
• Theorem 2. If one of two parallel lines is perpendicular to a plane, then the other line is also perpendicular to this plane.
• Theorem 3(about the parallelism of lines perpendicular to a plane). If two lines are perpendicular to the same plane, then they are parallel.
• Theorem 4(a sign of perpendicularity of a straight line and a plane). If a line is perpendicular to two intersecting lines lying in a plane, then it is perpendicular to this plane.
• Theorem 5(about a plane passing through a given point and perpendicular to a given line). Through any point in space there passes a single plane perpendicular to a given line.
• Theorem 6(about a line passing through a given point and perpendicular to a given plane). Through any point in space there passes a single straight line perpendicular to a given plane.
• Theorem 7(about the property of the diagonal rectangular parallelepiped). Square of the diagonal length of a cuboid equal to the sum squares of the lengths of its three edges having a common vertex:

Consequence: All four diagonals of a rectangular parallelepiped are equal to each other.

Three Perpendicular Theorem

Let the point A does not lie on a plane α . Let's draw through the point A straight line perpendicular to the plane α , and denote by the letter ABOUT the point of intersection of this line with the plane α . A perpendicular drawn from a point A to the plane α , called a segment JSC, dot ABOUT called the base of the perpendicular. If JSC– perpendicular to the plane α , A M– an arbitrary point of this plane, different from the point ABOUT, then the segment AM called an oblique drawn from a point A to the plane α , and point M– inclined base. Line segment OM– orthogonal projection (or, in short, projection) oblique AM to the plane α . Now we present a theorem that plays important role when solving many problems.

Theorem 1 (about three perpendiculars): A straight line drawn in a plane and perpendicular to the projection of an inclined one onto this plane is also perpendicular to the inclined one. The converse is also true:

Theorem 2 (about three perpendiculars): A straight line drawn in a plane and perpendicular to an inclined one is also perpendicular to its projection onto this plane. These theorems, for the notation from the drawing above, can be briefly formulated as follows:

Theorem: If from one point taken outside the plane, a perpendicular and two inclined ones are drawn to this plane, then:

• two obliques having equal projections are equal;
• Of the two inclined ones, the one whose projection is larger is greater.

Determination of distances by objects in space:

• The distance from a point to a plane is the length of a perpendicular drawn from this point to a given plane.
• The distance between parallel planes is the distance from an arbitrary point of one of the parallel planes to the other plane.
• The distance between a straight line and a plane parallel to it is the distance from an arbitrary point on the straight line to the plane.
• The distance between intersecting lines is the distance from one of the intersecting lines to a plane passing through another line and parallel to the first line.

Definition: In stereometry orthogonal projection straight a to the plane α the projection of this line onto the plane is called α if the straight line defining the design direction is perpendicular to the plane α .

Comment: As can be seen from the previous definition, there are many projections. Other (except orthogonal) projections of a line onto a plane can be constructed if the line defining the direction of projection is not perpendicular to the plane. However, it is the orthogonal projection of a line onto a plane that we will encounter in future problems. And we will simply call the orthogonal projection a projection (as in the drawing).

Definition: The angle between a straight line, not perpendicular to a plane, and this plane is the angle between a straight line and its orthogonal projection onto a given plane (angle AOA’ in the drawing above).

Theorem: The angle between a line and a plane is the smallest of all the angles that a given line forms with lines lying in a given plane and passing through the point of intersection of the line and the plane.

Definitions:

• Dihedral angle is a figure formed by two half-planes with a common boundary line and a part of space for which these half-planes serve as a boundary.
• Linear dihedral angle is an angle whose sides are rays with a common origin on the edge of a dihedral angle, which are drawn in its faces perpendicular to the edge.

Thus, the linear angle of a dihedral angle is the angle formed by the intersection of a dihedral angle with a plane perpendicular to its edge. All linear angles of a dihedral angle are equal to each other. The degree measure of a dihedral angle is the degree measure of its linear angle.

A dihedral angle is called right (acute, obtuse) if its degree measure is 90° (less than 90°, greater than 90°). In the future, when solving problems in stereometry, by a dihedral angle we will always understand that linear angle whose degree measure satisfies the condition:

Definitions:

• A dihedral angle at an edge of a polyhedron is a dihedral angle whose edge contains an edge of the polyhedron, and the faces of a dihedral angle contain faces of the polyhedron that intersect along a given edge of the polyhedron.
• The angle between intersecting planes is the angle between straight lines drawn respectively in these planes perpendicular to their line of intersection through a certain point.
• Two planes are called perpendicular if the angle between them is 90°.

Theorems:

• Theorem 1(a sign of perpendicularity of planes). If one of two planes passes through a line perpendicular to the other plane, then these planes are perpendicular.
• Theorem 2. A line lying in one of two perpendicular planes and perpendicular to the line along which they intersect is perpendicular to the other plane.

Symmetry of figures

Definitions:

1. Points M And M 1 are called symmetrical about the point O , If O is the midpoint of the segment MM 1 .
2. Points M And M 1 are called symmetrical about a straight line l , if straight l MM 1 and perpendicular to it.
3. Points M And M 1 are called symmetrical relative to the plane α , if the plane α passes through the middle of the segment MM 1 and perpendicular to this segment.
4. Dot O(straight l, plane α ) is called center (axis, plane) of symmetry figure, if each point of the figure is symmetrical relative to the point O(straight l, plane α ) at some point of the same figure.
5. A convex polyhedron is called correct, if all its faces are equal regular polygons and the same number of edges converge at each vertex.

Prism

Definitions:

1. Prism– a polyhedron, two of whose faces are equal polygons lying in parallel planes, and the remaining faces are parallelograms having common sides with these polygons.
2. Grounds – these are two faces that are equal polygons lying in parallel planes. In the drawing it is: ABCDE And KLMNP.
3. Side faces– all edges except the bases. Each side face is necessarily a parallelogram. In the drawing it is: ABLK, BCML, CDNM, DEPN And EAKP.
4. Side surface– union of side faces.
5. Full surface– combination of bases and side surface.
6. Side ribs– common sides of the side faces. In the drawing it is: A.K., B.L., C.M., DN And E.P..
7. Height– a segment connecting the bases of the prism and perpendicular to them. In the drawing this is, for example, KR.
8. Diagonal– a segment connecting two vertices of a prism that do not belong to the same face. In the drawing this is, for example, B.P..
9. Diagonal plane– a plane passing through the side edge of the prism and the diagonal of the base. Another definition: diagonal plane– a plane passing through two lateral edges of the prism that do not belong to the same face.
10. Diagonal section– intersection of a prism and a diagonal plane. A parallelogram is formed in the cross-section, including, sometimes, its special cases - rhombus, rectangle, square. In the drawing this is, for example, EBLP.
11. Perpendicular (orthogonal) section– the intersection of a prism and a plane perpendicular to its side edge.

Properties and formulas for a prism:

• The bases of the prism are equal polygons.
• The lateral faces of the prism are parallelograms.
• The lateral edges of the prism are parallel and equal.
• Prism volume equal to the product of its height and the area of ​​its base:

Where: S base – base area (in the drawing this is, for example, ABCDE), h– height (in the drawing this is MN).

• Total surface area of ​​the prism equal to the sum of the area of ​​its lateral surface and twice the area of ​​the base:

• The perpendicular section is perpendicular to all the side edges of the prism (in the drawing below, the perpendicular section is A 2 B 2 C 2 D 2 E 2).
• The angles of the perpendicular section are the linear angles of the dihedral angles with the corresponding lateral edges.
• A perpendicular (orthogonal) section is perpendicular to all side faces.
• Volume of an inclined prism equal to the product of the perpendicular cross-sectional area and the length of the lateral edge:

Where: S sec – perpendicular section area, l– length of the side rib (in the drawing below this is, for example, A.A. 1 or BB 1 and so on).

• Lateral surface area of an arbitrary prism is equal to the product of the perimeter of the perpendicular section and the length of the side edge:

Where: P sec – perimeter of a perpendicular section, l– length of the side rib.

Types of prisms in stereometry:

• If the side edges are not perpendicular to the base, then such a prism is called inclined(pictured above). The bases of such a prism, as usual, are located in parallel planes, the side ribs are not perpendicular to these planes, but parallel to each other. The lateral faces are parallelograms.
• - a prism in which all lateral edges are perpendicular to the base. In a straight prism, the lateral edges are the heights. The lateral faces of a straight prism are rectangles. And the area and perimeter of the base are equal, respectively, to the area and perimeter of the perpendicular section (in a straight prism, generally speaking, the perpendicular section is entirely the same figure as the base). Therefore, the area of ​​the lateral surface of a straight prism is equal to the product of the perimeter of the base and the length of the lateral edge (or, in in this case, prism height):

Where: P base – perimeter of the base of a straight prism, l– length of the side edge, equal to the height in a straight prism ( h). The volume of a straight prism is found by the general formula: V = S main ∙ h = S main ∙ l.

• Correct prism– a prism at the base of which lies a regular polygon (i.e. one in which all sides and all angles are equal to each other), and the side edges are perpendicular to the planes of the base. Examples of correct prisms:

Properties of a regular prism:

1. The bases of a regular prism are regular polygons.
2. The lateral faces of a regular prism are equal rectangles.
3. The lateral edges of a regular prism are equal to each other.
4. A correct prism is straight.

Definition: Parallelepiped – This is a prism whose bases are parallelograms. In this definition keyword is a "prism". Thus, a parallelepiped is a special case of a prism, which differs from the general case only in that at its base it does not have an arbitrary polygon, but rather a parallelogram. Therefore, all the above properties, formulas and definitions regarding a prism remain relevant for a parallelepiped. However, several additional properties characteristic of a parallelepiped can be identified.

Other properties and definitions:

• Two faces of a parallelepiped that do not have a common edge are called opposite, and having a common edge – adjacent.
• Two vertices of a parallelepiped that do not belong to the same face are called opposite.
• A line segment connecting opposite vertices is called diagonally parallelepiped.
• A parallelepiped has six faces and all of them are parallelograms.
• The opposite faces of a parallelepiped are equal and parallel in pairs.
• A parallelepiped has four diagonals; they all intersect at one point, and each of them is divided in half by this point.
• If the four lateral faces of a parallelepiped are rectangles (and the bases are arbitrary parallelograms), then it is called direct(in this case, like a straight prism, all the side edges are perpendicular to the bases). All properties and formulas for a straight prism are relevant for a right parallelepiped.
• The parallelepiped is called inclined, if not all of its side faces are rectangles.
• Volume of a straight or inclined parallelepiped is calculated using the general formula for the volume of the prism, i.e. equal to the product of the area of ​​the base of the parallelepiped and its height ( V = S main ∙ h).
• A right parallelepiped in which all six faces are rectangles (i.e., in addition to the side faces, the bases are also rectangles) is called rectangular. For a rectangular parallelepiped, all the properties of a right parallelepiped are relevant, as well as:
  • d and his ribs a, b, c are related by the relation:

d 2 = a 2 + b 2 + c 2 .

• • From the general formula for the volume of a prism, we can obtain the following formula for volume of a rectangular parallelepiped:

• A rectangular parallelepiped, all of whose faces are equal squares, is called cube. Among other things, the cube is a regular quadrangular prism, and in general a regular polyhedron. For a cube, all the properties of a rectangular parallelepiped and the properties of regular prisms are valid, as well as:
  • Absolutely all edges of the cube are equal to each other.
  • Diagonal of a cube d and the length of its edge a are related by the relation:

• From the formula for the volume of a rectangular parallelepiped, we can obtain the following formula for cube volume:

Pyramid

Definitions:

• Pyramid– a polyhedron, the base of which is a polygon, and the remaining faces are triangles having a common vertex. Based on the number of corners of the base, pyramids are classified as triangular, quadrangular, and so on. The figure shows examples: quadrangular and hexagonal pyramids.

• Base– a polygon that does not belong to the vertex of the pyramid. In the drawing the base is BCDE.
• Faces other than the base are called lateral. In the drawing it is: ABC, ACD, ADE And AEB.
• The common vertex of the lateral faces is called the top of the pyramid(precisely the top of the entire pyramid, and not just the top, like all the other tops). In the drawing it is A.
• The edges connecting the top of the pyramid with the vertices of the base are called lateral. In the drawing it is: AB, A.C., AD And A.E..
• When designating a pyramid, first name its apex, and then the apexes of the base. For the pyramid from the drawing the designation will be as follows: ABCDE.

• Heightpyramids is called a perpendicular drawn from the top of the pyramid to its base. The length of this perpendicular is indicated by the letter H. In the drawing the height is A.G.. Note: only if the pyramid is a regular quadrangular pyramid (as in the drawing) does the height of the pyramid fall on the diagonal of the base. In other cases this is not the case. In general, for an arbitrary pyramid, the intersection point of the height and the base can be anywhere.
• Apothem – side edge height correct pyramid drawn from its top. In the drawing this is, for example, A.F..
• Diagonal section of a pyramid- a section of a pyramid passing through the top of the pyramid and the diagonal of the base. In the drawing this is, for example, ACE.

Another stereometric drawing with symbols for better memorization(the picture shows a regular triangular pyramid):

If all lateral edges ( S.A., S.B., S.C., SD in the drawing below) the pyramids are equal, then:

• A circle can be described around the base of the pyramid, with the top of the pyramid projected to its center (point O). In other words, height (segment SO), lowered from the top of such a pyramid to the base ( ABCD), falls into the center of the circle described around the base, i.e. at the point of intersection of the bisectoral perpendiculars of the base.
• The side ribs form equal angles with the plane of the base (in the drawing below these are the angles SAO, SBO, SCO, S.D.O.).

Important: The opposite is also true, that is, if the side edges form equal angles with the plane of the base, or if a circle can be described around the base of the pyramid, with the top of the pyramid projected into its center, then all the side edges of the pyramid are equal.

If the side faces are inclined to the base plane at one angle (angles DMN, DKN, DLN in the drawing below are equal), then:

• A circle can be inscribed at the base of the pyramid, and the top of the pyramid is projected into its center (point N). In other words, height (segment DN), lowered from the top of such a pyramid to the base, falls into the center of the circle inscribed in the base, i.e. at the point of intersection of the bisectors of the base.
• The heights of the side faces (apothem) are equal. In the drawing below DK, D.L., DM– equal apothems.
• The lateral surface area of ​​such a pyramid equal to half the product of the perimeter of the base and the height of the side face (apothem).

Where: P– perimeter of the base, a– length of apothem.

Important: The opposite is also true, that is, if a circle can be inscribed at the base of the pyramid, and the top of the pyramid is projected into its center, then all the side faces are inclined to the plane of the base at the same angle and the heights of the side faces (apothem) are equal.

Correct pyramid

Definition: The pyramid is called correct, if its base is a regular polygon, and its vertex is projected to the center of the base. Then it has the following properties:

• All lateral edges of a regular pyramid are equal.
• All lateral faces of a regular pyramid are inclined to the plane of the base at the same angle.

Important Note: As you can see, regular pyramids are one of those pyramids that include the properties outlined just above. Indeed, if the base of a regular pyramid is a regular polygon, then the center of its inscribed and circumscribed circles coincide, and the vertex of a regular pyramid is projected precisely into this center (by definition). However, it is important to understand that not only correct pyramids may have the properties discussed above.

• In a regular pyramid, all lateral faces are equal isosceles triangles.
• You can either fit a sphere into any regular pyramid or describe a sphere around it.
• The area of ​​the lateral surface of a regular pyramid is equal to half the product of the perimeter of the base and the apothem.

Formulas for volume and area of ​​a pyramid

Theorem(about the volume of pyramids having equal heights And equal areas grounds). Two pyramids that have equal heights and equal base areas have equal volumes (Of course, you probably already know the formula for the volume of a pyramid, or you see it a few lines below, and this statement seems obvious to you, but in fact, if you judge eye", then this theorem is not so obvious (see figure below). This also applies, by the way, to other polyhedra and geometric shapes: their appearance is deceptive, therefore, really - in mathematics you need to trust only formulas and correct calculations).

• Volume of the pyramid can be calculated using the formula:

Where: S base – area of ​​the base of the pyramid, h– height of the pyramid.

• Lateral surface of the pyramid equal to the sum of the areas of the lateral faces. For the area of ​​the lateral surface of a pyramid, we can formally write the following stereometric formula:

Where: S side – lateral surface area, S 1 , S 2 , S 3 – areas of the side faces.

• Full surface of the pyramid equal to the sum of the lateral surface area and the base area:

Definitions:

• - the simplest polyhedron, the faces of which are four triangles, in other words, a triangular pyramid. For a tetrahedron, any of its faces can serve as a base. In total, a tetrahedron has 4 faces, 4 vertices and 6 edges.
• The tetrahedron is called correct, if all its faces are equilateral triangles. For a regular tetrahedron:
  1. All edges of a regular tetrahedron are equal to each other.
  2. All faces of a regular tetrahedron are equal to each other.
  3. The perimeters, areas, heights and all other elements of all faces are respectively equal to each other.

The drawing shows a regular tetrahedron, with triangles ABC, ADC, CBD, BAD- equal. From general formulas for the volume and area of ​​a pyramid, as well as knowledge from planimetry, it is not difficult to obtain formulas for volume and area of ​​a regular tetrahedron(A– rib length):

Definition: When solving problems in stereometry, a pyramid is called rectangular, if one of the side edges of the pyramid is perpendicular to the base. In this case, this edge is the height of the pyramid. Below are examples of triangular and pentagonal rectangular pyramids. In the picture on the left S.A.– an edge that is also a height.

Truncated pyramid

Definitions and properties:

• Truncated pyramid is called a polyhedron enclosed between the base of the pyramid and a cutting plane parallel to its base.
• The figure obtained at the intersection of the cutting plane and the original pyramid is also called basis truncated pyramid. So, the truncated pyramid in the drawing has two bases: ABC And A 1 B 1 C 1 .
• The lateral faces of the truncated pyramid are trapezoids. In the drawing this is, for example, A.A. 1 B 1B.
• The lateral ribs of a truncated pyramid are the parts of the ribs of the original pyramid enclosed between the bases. In the drawing this is, for example, A.A. 1 .
• The height of a truncated pyramid is a perpendicular (or the length of this perpendicular) drawn from some point in the plane of one base to the plane of another base.
• A truncated pyramid is called correct, if it is a polyhedron that is cut off by a plane parallel to the base correct pyramids.
• The bases of a regular truncated pyramid are regular polygons.
• The lateral faces of a regular truncated pyramid are isosceles trapezoids.
• Apotheme of a regular truncated pyramid is the height of its side face.
• The lateral surface area of ​​a truncated pyramid is the sum of the areas of all its lateral faces.

Formulas for a truncated pyramid

The volume of a truncated pyramid is equal to:

Where: S 1 and S 2 – base area, h– the height of the truncated pyramid. However, in practice, it is more convenient to search for the volume of a truncated pyramid this way: you can build a truncated pyramid into a pyramid by extending the side ribs until they intersect. Then the volume of the truncated pyramid can be found as the difference between the volumes of the entire pyramid and the completed part. The lateral surface area can also be sought as the difference between the lateral surface areas of the entire pyramid and the completed part. Lateral surface area of ​​a regular truncated pyramid is equal to the half-product of the sum of the perimeters of its bases and the apothem:

Where: P 1 and P 2 – perimeters of the bases correct truncated pyramid, A– length of apothem. The total surface area of ​​any truncated pyramid is obviously found as the sum of the areas of the bases and the lateral surface:

Pyramid and ball (sphere)

Theorem: Near the pyramid you can describe the area when at the base of the pyramid lies an inscribed polygon (i.e. a polygon around which a sphere can be described). This condition is necessary and sufficient. The center of the sphere will be the intersection point of the planes passing through the midpoints of the edges of the pyramid perpendicular to them.

Note: From this theorem it follows that a sphere can be described both around any triangular and around any regular pyramid. However, the list of pyramids around which the sphere can be described is not limited to these types of pyramids. In the drawing on the right, at height SH need to choose a point ABOUT, equidistant from all vertices of the pyramid: SO = OB = OS = O.D. = O.A.. Then point ABOUT– the center of the circumscribed sphere.

Theorem: You can go into the pyramid enter the sphere when the bisector planes of the internal dihedral angles of the pyramid intersect at one point (a necessary and sufficient condition). This point will be the center of the sphere.

Comment: You obviously did not understand what you read in the line above. However, the main thing to remember is that any regular pyramid is one into which a sphere can be inscribed. Moreover, the list of pyramids into which the sphere can be entered is not limited to the correct ones.

Definition: Bisector plane divides a dihedral angle in half, and each point of the bisector plane is equidistant from the faces forming the dihedral angle. In the figure on the right the plane γ is the bisector plane of the dihedral angle formed by the planes α And β .

The stereometric drawing below shows a ball inscribed in a pyramid (or a pyramid described around a ball), while the point ABOUT– the center of the inscribed sphere. This point ABOUT equidistant from all faces of the ball, for example:

OM = OO 1

Pyramid and cone

In stereometry a cone is said to be inscribed in a pyramid, if their vertices coincide and its base is inscribed in the base of the pyramid. Moreover, it is possible to fit a cone into a pyramid only when the apothems of the pyramid are equal to each other (a necessary and sufficient condition).

The cone is called circumscribed about the pyramid, when their vertices coincide, and its base is described near the base of the pyramid. Moreover, it is possible to describe a cone near a pyramid only when all the lateral edges of the pyramid are equal to each other (a necessary and sufficient condition).

Important property:

Pyramid and cylinder

A cylinder is said to be inscribed in a pyramid, if one of its bases coincides with the circle inscribed in the section of the pyramid by a plane parallel to the base, and the other base belongs to the base of the pyramid.

The cylinder is called circumscribed about the pyramid, if the top of the pyramid belongs to one of its bases, and its other base is described near the base of the pyramid. Moreover, it is possible to describe a cylinder near a pyramid only if there is an inscribed polygon at the base of the pyramid (a necessary and sufficient condition).

Sphere and ball

Definitions:

1. Sphere– a closed surface, the geometric locus of points in space equidistant from a given point, called center of the sphere. A sphere is also a body of revolution, formed by rotating a semicircle around its diameter. Radius of the sphere called a segment connecting the center of the sphere with any point on the sphere.
2. Chordoy sphere is a segment connecting two points on the sphere.
3. Diameter of a sphere is called a chord passing through its center. The center of the sphere divides any of its diameters into two equal segments. Any diameter of a sphere with radius R equals 2 R.
4. Ball– geometric body; the collection of all points in space that are located at a distance no greater than a given one from some center. This distance is called radius of the ball. The ball is formed by rotating a semicircle around its fixed diameter. Note: the surface (or boundary) of a ball is called a sphere. We can also give the following definition of a ball: a ball is a geometric body consisting of a sphere and part of the space limited by this sphere.
5. Radius, chord And diameter of a ball are called the radius, chord and diameter of the sphere, which is the boundary of this ball.
6. The difference between a ball and a sphere is similar to the difference between a circle and a circle. A circle is a line, and a circle is also all the points inside this line. A sphere is a shell, and a ball is also all the points inside this shell.
7. The plane passing through the center of the sphere (ball) is called center plane.
8. The section of a sphere (ball) by the diametrical plane is called large circle (big circle).

Theorems:

• Theorem 1(about the section of a sphere by a plane). The section of a sphere by a plane is a circle. Note that the theorem remains true even if the plane passes through the center of the sphere.
• Theorem 2(about the section of a ball by a plane). The section of a ball by a plane is a circle, and the base of the perpendicular drawn from the center of the ball to the plane of the section is the center of the circle obtained in the section.

The largest circle that can be obtained in a section of a given ball by a plane lies in a section passing through the center of the ball ABOUT. This is what is called the great circle. Its radius is equal to the radius of the ball. Any two large circles intersect along the diameter of the ball AB. This diameter is also the diameter of intersecting great circles. Through two points of the spherical surface located at the ends of the same diameter (in Fig. A And B), you can draw countless large circles. For example, an infinite number of meridians can be drawn through the Earth's poles.

Definitions:

1. Tangent plane to the sphere is a plane that has only one common point with a sphere, and their common point is called the point of tangency between the plane and the sphere.
2. Tangent plane to the ball is called the tangent plane to the sphere, which is the boundary of this ball.
3. Any straight line lying in the tangent plane of a sphere (ball) and passing through the point of contact is called tangent line to the sphere (ball). By definition, a tangent plane has only one common point with a sphere, therefore, a tangent line also has only one common point with a sphere - the point of tangency.

Theorems:

• Theorem 1(sign of a tangent plane to a sphere). A plane perpendicular to the radius of the sphere and passing through its end lying on the sphere touches the sphere.
• Theorem 2(about the property of a tangent plane to a sphere). The tangent plane to the sphere is perpendicular to the radius drawn to the point of tangency.

Polyhedra and sphere

Definition: In stereometry, a polyhedron (for example, a pyramid or a prism) is called included in the sphere, if all its vertices lie on the sphere. In this case, the sphere is said to be circumscribed about a polyhedron (pyramid, prism). Similarly: a polyhedron is called inscribed in a ball, if all its vertices lie on the boundary of this ball. In this case, the ball is said to be circumscribed about the polyhedron.

Important property: The center of a sphere circumscribed about a polyhedron is located at a distance equal to the radius R spheres, from each vertex of the polyhedron. Here are examples of polyhedra inscribed in a sphere:

Definition: The polyhedron is called described around a sphere (ball), if the sphere (ball) touches everyone polyhedron faces. In this case, the sphere and the ball are said to be inscribed in the polyhedron.

Important: The center of a sphere inscribed in a polyhedron is located at a distance equal to the radius r spheres, from each of the planes containing the faces of the polyhedron. Here are examples of polyhedra described near a sphere:

Volume and surface area of ​​a sphere

Theorems:

• Theorem 1(about the area of ​​a sphere). The area of ​​the sphere is:

Where: R– radius of the sphere.

• Theorem 2(about the volume of the ball). Volume of a sphere with radius R calculated by the formula:

Ball segment, layer, sector

In stereometry ball segment The part of the ball cut off by the cutting plane is called. In this case, the relationship between the height, the radius of the base of the segment and the radius of the ball:

Where: h− segment height, r− radius of the segment base, R− radius of the ball. Base area of ​​the ball segment:

Outer surface area of ​​the ball segment:

Total surface area of ​​the ball segment:

Ball segment volume:

In stereometry spherical layer is the part of a sphere enclosed between two parallel planes. Outer surface area of ​​the spherical layer:

Where: h− height of the spherical layer, R− radius of the ball. Total surface area of ​​the spherical layer:

Where: h− height of the spherical layer, R− radius of the ball, r 1 , r 2 – radii of the bases of the spherical layer, S 1 , S 2 – areas of these bases. The easiest way to find the volume of a spherical layer is as the difference in the volumes of two spherical segments.

In stereometry spherical sector called a part of a ball consisting of a spherical segment and a cone with the vertex in the center of the ball and the base coinciding with the base of the spherical segment. This implies that the spherical segment is less than half a ball. Total surface area of ​​the spherical sector:

Where: h− height of the corresponding spherical segment, r− radius of the base of the spherical segment (or cone), R− radius of the ball. The volume of the spherical sector is calculated by the formula:

Definitions:

1. In a certain plane, consider a circle with center O and radius R. Through each point of the circle we draw a straight line perpendicular to the plane of the circle. Cylindrical surface the figure formed by these lines is called, and the lines themselves are called forming a cylindrical surface. All generatrices of a cylindrical surface are parallel to each other, since they are perpendicular to the plane of the circle.

1. Straight circular cylinder or simply cylinder is a geometric body bounded by a cylindrical surface and two parallel planes that are perpendicular to the generatrices of the cylindrical surface. Informally, you can think of a cylinder as a straight prism with a circle at its base. This will help you easily understand and, if necessary, derive formulas for the volume and lateral surface area of ​​the cylinder.
2. Lateral surface of the cylinder is called a part of a cylindrical surface located between secant planes that are perpendicular to its generators, and parts (circles) cut off by a cylindrical surface on parallel planes are called cylinder bases. The bases of the cylinder are two equal circles.
3. Generator of the cylinder called a segment (or the length of this segment) of the generatrix of a cylindrical surface located between the parallel planes in which the bases of the cylinder lie. All generatrices of the cylinder are parallel and equal to each other, and also perpendicular to the bases.
4. Cylinder axis called a segment connecting the centers of the circles that are the bases of the cylinder.
5. Cylinder height is called a perpendicular (or the length of this perpendicular) drawn from some point in the plane of one base of the cylinder to the plane of another base. In a cylinder, the height is equal to the generator.
6. Cylinder radius is called the radius of its bases.
7. The cylinder is called equilateral, if its height is equal to the diameter of the base.
8. A cylinder can be obtained by rotating a rectangle around one of its sides by 360°.
9. If the secant plane is parallel to the cylinder axis, then the section of the cylinder is a rectangle, two sides of which are generatrices, and the other two are chords of the bases of the cylinder.
10. Axial section A section of a cylinder is called a section of the cylinder by a plane passing through its axis. The axial section of the cylinder is a rectangle, two sides of which are the generators of the cylinder, and the other two are the diameters of its bases.
11. If the cutting plane is perpendicular to the axis of the cylinder, then a circle equal to the bases is formed in the section. In the drawing below: on the left - axial section; in the center - a section parallel to the cylinder axis; on the right is a section parallel to the base of the cylinder.

Cylinder and prism

A prism is said to be inscribed in a cylinder, if its bases are inscribed in the bases of the cylinder. In this case, the cylinder is said to be circumscribed about the prism. The height of the prism and the height of the cylinder in this case will be equal. All lateral edges of the prism will belong to the lateral surface of the cylinder and coincide with its generatrices. Since by cylinder we mean only a straight cylinder, then only a straight prism can be inscribed into such a cylinder. Examples:

The prism is called circumscribed about the cylinder, if its bases are described near the bases of the cylinder. In this case, the cylinder is said to be inscribed in a prism. The height of the prism and the height of the cylinder in this case will also be equal. All side edges of the prism will be parallel to the generatrices of the cylinder. Since by cylinder we mean only a straight cylinder, then such a cylinder can only be inscribed in a straight prism. Examples:

Cylinder and sphere

A sphere (ball) is said to be inscribed in a cylinder, if it touches the bases of the cylinder and each of its generatrices. In this case, the cylinder is said to be circumscribed about a sphere (ball). A sphere can be inscribed in a cylinder only if it is an equilateral cylinder, i.e. the diameter of its base and height are equal to each other. The center of the inscribed sphere will be the middle of the cylinder axis, and the radius of this sphere will coincide with the radius of the cylinder. Example:

A cylinder is said to be inscribed in a sphere, if the circles of the bases of the cylinder are sections of a sphere. A cylinder is said to be inscribed in a sphere if the bases of the cylinder are sections of the sphere. In this case, the ball (sphere) is called circumscribed about the cylinder. A sphere can be described around any cylinder. The center of the described sphere will also be the middle of the cylinder axis. Example:

Based on the Pythagorean theorem, it is easy to prove the following formula relating the radius of the circumscribed sphere ( R), cylinder height ( h) and cylinder radius ( r):

Volume and area of ​​the lateral and full surfaces of the cylinder

Theorem 1(about the area of ​​the lateral surface of a cylinder): The area of ​​the lateral surface of a cylinder is equal to the product of the circumference of its base and its height:

Where: R– radius of the cylinder base, h- his high. This formula is easily derived (or proven) based on the formula for the lateral surface area of ​​a straight prism.

Total surface area of ​​the cylinder, as usual in stereometry, is the sum of the areas of the lateral surface and the two bases. The area of ​​each base of the cylinder (i.e. simply the area of ​​the circle) is calculated using the formula:

Therefore, the total surface area of ​​the cylinder is S full cylinder is calculated by the formula:

Theorem 2(about the volume of a cylinder): The volume of a cylinder is equal to the product of the area of ​​the base and the height:

Where: R And h are the radius and height of the cylinder, respectively. This formula is also easily derived (proved) based on the formula for the volume of a prism.

Theorem 3(Archimedes): The volume of a sphere is one and a half times less than the volume of the cylinder circumscribed around it, and the surface area of ​​such a sphere is one and a half times less than the total surface area of ​​the same cylinder:

Cone

Definitions:

1. A cone (more precisely, a circular cone) called a body that consists of a circle (called base of the cone), point not lying in the plane of this circle (called the top of the cone) and all possible segments connecting the vertex of the cone with the points of the base. Informally, you can think of a cone as a regular pyramid with a circle at its base. This will help you easily understand and, if necessary, derive formulas for the volume and area of ​​the lateral surface of a cone.

1. The segments (or their lengths) connecting the vertex of the cone with the points of the base circle are called forming a cone. All generators of a right circular cone are equal to each other.
2. The surface of the cone consists of the base of the cone (circle) and the side surface (composed of all possible generatrices).
3. The union of the generators of a cone is called generating (or lateral) surface of the cone. The forming surface of the cone is a conical surface.
4. The cone is called direct, if the straight line connecting the top of the cone with the center of the base is perpendicular to the plane of the base. In what follows, we will consider only the straight cone, calling it simply a cone for brevity.
5. A straight circular cone can be visually imagined as a body obtained by rotation right triangle around its leg as an axis. In this case, the lateral surface of the cone is formed by the rotation of the hypotenuse, and the base is formed by the rotation of the leg, which is not an axis.
6. Cone radius is called the radius of its base.
7. Cone height called the perpendicular (or its length) dropped from its vertex to the plane of the base. For a straight cone, the base of the height coincides with the center of the base. The axis of a right circular cone is a straight line containing its height, i.e. a straight line passing through the center of the base and the top.
8. If the secant plane passes through the axis of the cone, then the section is an isosceles triangle, the base of which is the diameter of the base of the cone, and the sides are the generatrices of the cone. This section is called axial.

1. If the cutting plane passes through the internal point of the height of the cone and is perpendicular to it, then the section of the cone is a circle, the center of which is the point of intersection of the height and this plane.
2. Height ( h), radius ( R) and the length of the generatrix ( l) of a right circular cone satisfy the obvious relation:

Volume and area of ​​the lateral and full surfaces of the cone

Theorem 1(about the area of ​​the lateral surface of the cone). The area of ​​the lateral surface of the cone is equal to the product of half the circumference of the base and the generatrix:

Where: R– radius of the cone base, l– length of the cone generatrix. This formula is easily derived (or proven) based on the formula for the lateral surface area of ​​a regular pyramid.

Total surface area of ​​the cone is called the sum of the lateral surface area and the base area. The area of ​​the base of the cone (i.e. simply the area of ​​the circle) is equal to: S basic = πR 2. Therefore, the total surface area of ​​the cone is S full cone is calculated by the formula:

Theorem 2(about the volume of a cone). The volume of a cone is equal to one third of the product of the area of ​​the base and the height:

Where: R– radius of the cone base, h- his high. This formula is also easily derived (proven) based on the formula for the volume of the pyramid.

Definitions:

1. A plane parallel to the base of the cone and intersecting the cone cuts off a smaller cone from it. The remaining part is called truncated cone.

1. The base of the original cone and the circle resulting from the section of this cone by a plane are called reasons, and the segment connecting their centers is truncated cone height.
2. A straight line passing through the height of a truncated cone (i.e. through the centers of its bases) is its axis.
3. The part of the lateral surface of the cone that limits the truncated cone is called it lateral surface, and the segments of the cone’s generatrices located between the bases of the truncated cone are called its forming.
4. All generators of a truncated cone are equal to each other.
5. A truncated cone can be obtained by rotating 360° rectangular trapezoid around its side, perpendicular to the bases.

Formulas for a truncated cone:

The volume of a truncated cone is equal to the difference between the volumes of the full cone and the cone cut off by a plane parallel to the base of the cone. The volume of a truncated cone is calculated by the formula:

Where: S 1 = π r 1 2 and S 2 = π r 2 2 – area of ​​the bases, h– height of the truncated cone, r 1 and r 2 – radii of the upper and lower bases of the truncated cone. However, in practice, it is still more convenient to search for the volume of a truncated cone as the difference between the volumes of the original cone and the cut-off part. The lateral surface area of ​​a truncated cone can also be sought as the difference between the lateral surface areas of the original cone and the cut-off part.

Indeed, the area of ​​the lateral surface of a truncated cone is equal to the difference between the areas of the lateral surfaces of the full cone and the cone cut off by a plane parallel to the base of the cone. Lateral surface area of ​​a truncated cone calculated by the formula:

Where: P 1 = 2π r 1 and P 2 = 2π r 2 – perimeters of the bases of a truncated cone, l– length of the generatrix. Total surface area of ​​a truncated cone, obviously, is found as the sum of the areas of the bases and the lateral surface:

Please note that the formulas for the volume and lateral surface area of ​​a truncated cone are derived from formulas for similar characteristics of a regular truncated pyramid.

Cone and sphere

A cone is said to be inscribed in a sphere(ball), if its vertex belongs to the sphere (the boundary of the ball), and the circumference of the base (the base itself) is a section of the sphere (ball). In this case, the sphere (ball) is said to be circumscribed about the cone. A sphere can always be described around a right circular cone. The center of the circumscribed sphere will lie on the straight line containing the height of the cone, and the radius of this sphere will be equal to the radius of the circle circumscribed about the axial section of the cone (this section is isosceles triangle). Examples:

A sphere (ball) is said to be inscribed in a cone, if the sphere (ball) touches the base of the cone and each of its generatrices. In this case, the cone is said to be circumscribed about a sphere (ball). You can always fit a sphere into a right circular cone. Its center will lie at the height of the cone, and the radius of the inscribed sphere will be equal to the radius of the circle inscribed in the axial section of the cone (this section is an isosceles triangle). Examples:

Cone and pyramid

• A cone is said to be inscribed in a pyramid (a pyramid is described around a cone) if the base of the cone is inscribed in the base of the pyramid, and the vertices of the cone and the pyramid coincide.
• A pyramid is called inscribed in a cone (a cone is described around a pyramid) if its base is inscribed in the base of the cone, and the lateral edges form the cone.
• The heights of such cones and pyramids are equal to each other.

Note: More details about how in stereometry a cone fits into a pyramid or is described around a pyramid have already been discussed in

How to successfully prepare for the CT in physics and mathematics?

In order to successfully prepare for the CT in physics and mathematics, among other things, it is necessary to fulfill three most important conditions:

1. Study all topics and complete all tests and assignments given in the educational materials on this site. To do this, you need nothing at all, namely: devote three to four hours every day to preparing for the CT in physics and mathematics, studying theory and solving problems. The fact is that CT is an exam where it is not enough just to know physics or mathematics, you also need to be able to solve it quickly and without failures a large number of tasks for different topics and of varying complexity. The latter can only be learned by solving thousands of problems.
2. Learn all the formulas and laws in physics, and formulas and methods in mathematics. In fact, this is also very simple to do; there are only about 200 necessary formulas in physics, and even a little less in mathematics. Each of these subjects has about a dozen standard methods for solving problems basic level difficulties that can also be learned, and thus solved completely automatically and without difficulty at the right time most CT. After this, you will only have to think about the most difficult tasks.
3. Attend all three stages of rehearsal testing in physics and mathematics. Each RT can be visited twice to decide on both options. Again, on the CT, in addition to the ability to quickly and efficiently solve problems, and knowledge of formulas and methods, you must also be able to properly plan time, distribute forces, and most importantly, correctly fill out the answer form, without confusing the numbers of answers and problems, or your own last name. Also, during RT, it is important to get used to the style of asking questions in problems, which may seem very unusual to an unprepared person at the DT.

Successful, diligent and responsible implementation of these three points will allow you to show an excellent result at the CT, the maximum of what you are capable of.

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If you think you have found an error in the training materials, please write about it by email. You can also report an error on the social network (). In the letter, indicate the subject (physics or mathematics), the name or number of the topic or test, the number of the problem, or the place in the text (page) where, in your opinion, there is an error. Also describe what the suspected error is. Your letter will not go unnoticed, the error will either be corrected, or you will be explained why it is not an error.

Name: Hrithik Roshan

Age: 45 years

Height: 178

Activity: film actor, director

Family status: divorced

Hrithik Roshan: biography

Hrithik Roshan is one of the the most beautiful actors India. The artist is also known as a director. The Bollywood star became widely famous and loved both in his homeland and far beyond the country's borders. The young actor’s track record already includes more than two dozen films. In addition, Hrithik is considered a sex symbol in his native country.

Childhood and youth

Hrithik Roshan was born on January 10, 1974. According to the zodiac sign of the young man, he is Capricorn. Roshan appeared in a family associated with cinema, which influenced his fate young man and choice of profession. Hrithik's father is Indian actor and director Rakesh Roshan, Pinky's mother is an actress, and his uncle Rajer is a famous Indian composer. However, for a long time, Hrithik was afraid to think about a career in cinema, since he had speech problems until the age of 14 and was born with a sixth finger on his hand.

However, at the age of six, in 1980, the boy first played in the film “Aasha”, then in the film “Aap Ke Deewane”. The continuation of Hrithik’s filmography followed only in 2000 in his father’s film “Say You Love”. For this role, the young actor received Filmfare Awards in the categories for “Best Actor” and “Best Debut”.

Hrithik's parents wanted him to get an education in Europe, and the guy himself secretly enrolled in courses acting. The father reacted to this in a peculiar way. He suggested that his son start a career in film production rather than as an actor. So future star became only his father's third assistant. He had to travel around the country, spending the night in cheap hotels, working his ass off.

Movies

Having learned the secrets of Bollywood from the inside, and not in dry theory, Hrithik finally gets the role that instantly made the guy a famous and famous actor. The young man’s father was looking for a performer to play the main role in his next film (“Say that you love!”). The actor he originally planned to direct turned down the role, and Rakesh decided to try his son. In addition, Rakesh wanted to bring a new hero into Indian films.

The man's calculation turned out to be correct. Hrithik became an overnight star. The young actor matured, and classes with a personal bodybuilding trainer, combined with a sports balanced diet, turned the artist into a sex symbol, for whom people went crazy female part India. Hrithik's sad green eyes and athletic pumped up figure have become the subject of dreams for many Indian women.

On the wave of success, Hrithik signs contracts for the next two years. However, new projects did not bring to the young actor planned success. Moreover, they created the impression that Hrithik is only able to work fruitfully together with his father. And it was he who came to the rescue of his son, giving him a chance to restore glory, and offered him excellent roles.

The film “You Are Not Alone” again elevates the actor to the Bollywood Olympus. The sci-fi film won a number of prestigious awards and was recognized as the best film of the year. Hrithik managed to consolidate his success in the blockbuster “Krrish” (2006), the role in which was named best job actor.

In the same year, the blockbuster "Bikers 2: True Feelings" brought him fame and popularity abroad. In this film, the artist worked with, Uday Chopra and others.

Hrithik played the role of Aryan Singh, a young man with the ability to transform. At the same time, the guy robs the main treasures of the world. Officer Jai Dixit tries to catch the criminal, but he is always one step ahead. Then the man comes up with a trick: Jai sends the burglar Sunehri to Aryan, who must ingratiate herself with the robber and then hand the guy over to the police.

Taught by bitter experience, the actor chooses new projects more carefully. This practice has borne fruit in the films “Jodha and Akbar” (2008), “Chance for Luck” (2009), “ Kites" and "Plea" in 2009 cemented Hrithik Roshan's name in Indian cinema. The film "Krrish" was continued in two parts.

In 2011, Hrithik appeared in the lead role in the film Life Can't Be Boring.

This is a story about three friends who go on vacation. On vacation, the lives of the young people change: one of them, Kabir, meets a girl, Natasha, with whom he becomes engaged six months later. Then the guy organizes a three-week bachelor party with his two best friends.

Along with Roshan, Naseeruddin Shah, Farhan Akhtar and others were involved in the film.

Then Hrithik starred in the film “Fire Path”. Central female character performed Indian actress.

A year later, the artist appeared on screen again with Katrina Kaif. The artists played in the action movie “Bang Bang.”

In the story, a modest bank employee, Harleen Sahni, meets a mysterious guy, Rajveer Nanda. This meeting leads to chases, shootouts and explosions. At the same time, the man assures the girl that he is a good person.

Along with Roshan and Kaifa, the film stars Pawan Malhotra, Danny Denzongpa, Javed Jaffrey and others.

2016 was marked by the successful release of the film “Mohenjo Daro”.

The film takes place in ancient city Mohenjo Daro. In the center of the plot are several residents who, in the background historical events live ordinary life and face everyday problems: falling in love, quarreling, getting married and breaking up.

Hrithik Roshan's colleagues on the set were Pooja Hegde, Kabir Bedi, Arunoday Sinkh and others.

According to the results of a survey of the audience in India, Hrithik was recognized as the best actor ever to portray in Bollywood historical character(Akbar). In this project he again starred with Aishwarya Rai.

Today, the artist is one of the most sought-after actors in India. Films with Hrithik and photos can be found on the Internet. Clips with dances from the most successful films enjoy constant success, gaining multi-million views. And music and songs from the tapes are among the most popular and downloaded on the Internet.

Personal life

The new millennium was marked for Hrithik with his marriage to Suzanne Khan, the daughter of the famous Indian director. The actor dated his future wife for 4 years. Having lived together for 13 years, a week before their next wedding anniversary, the couple divorced on December 20, 2013, stirring up the world of Indian show business and the actor’s many fans.

The former spouses have two children growing up - boys Rehan and Ridan. The father is actively involved in raising the children. At the same time, he sees his ex-wife only because of meetings with his sons. There were rumors that Suzanne was dating Hrithik's friend, but they were not confirmed. The former father-in-law spoke about a possible reunion of the couple, but this did not happen.

Later in the interview, Hrithik shared that he has a great relationship with Sussanne. The former spouses remained friends.

The news of the divorce revived rumors about Hrithik's possible romance with, which arose after the actors worked together in a number of projects. The public felt that the chemistry demonstrated on screen could not only be a talented performance and continued to exist in life.

Another contender for the heart of the handsome actor was Priyanka Chopra, with whom they worked together in the film “Krrish” in 2006. However, the rumors were not confirmed, and today the actor is free. A handsome, athletic man with a height of 182 centimeters and a weight of 74-78 kg wins women’s hearts both in his native India and around the world.

An audience of millions watches the famous Indian artist on microblogs

Among Bollywood stars, actor Hrithik Roshan is undoubtedly one of the most beloved by the audience and rightfully occupies the podium. Fans are watching with trepidation the new roles and films of the hero of our article. AND personal life of Hrithik Roshan Naturally, they also do not leave without due attention. Fortunately, the superstar is incredible open man, ready to communicate with both journalists and fans.

The biography of Hrithik Roshan is full of both happy and tragic events. Over the past 42 years (which is exactly how long he celebrated just the other day, on January 10), the actor had to overcome long haul from a shy Indian guy to a celebrated superstar. Despite the fact that Hrithik Roshan was born into a family of famous artists in India (his grandfather is a famous composer, and his father is a director), the physical disabilities that nature gave him (to right hand actor - six fingers, and until the age of 14 he suffered from speech problems), willpower and determination made him an idol of millions. To date, Hrithik Roshan’s biography includes about four dozen films, almost all of which have received recognition among both viewers and critics. The actor’s work is characterized by a desire for perfection, for which he is ready to make almost any sacrifice. For example, now he is forced to move with the help of crutches due to another injury on his film set. Apart from cinema, Hrithik Roshan's life is of great importance. own business(he is one of the owners of the clothing company HRX) and charity. Last year, the actor took the Paralympic athletes under his wing to allow them to prepare well and perform well at this year's Olympics. Although most often the celebrity tries not to advertise his actions related to charity.

In the photo - Hrithik Roshan and his ex-wife Sussanne Khan

In the photo - Hrithik Roshan with his sons

Hrithik Roshan's personal life is not going very smoothly. The whole country knows that he had a long 13-year marriage with Suzanne Khan, an interior designer and businesswoman. Their relationship seemed absolutely cloudless from the outside. Before the wedding, the couple dated for four years. His wife gave Hrithik Roshan two sons - Hrehan and Hridhan. Unfortunately, on December 13, 2013, the actor officially announced his divorce. Parting with his family was a huge shock for the celebrity. As of today, he is still single. Free time Hrithik Roshan prefers to spend time with his family, with his sons or with friends. By her own admission, he maintains relations with his ex-wife only at the level of raising children. This situation suits both of them quite well. Naturally, the press continues to monitor the lives of both, and therefore sometimes articles appear that become a source of gossip and frustration for their subjects. So, for example, last year Suzanne was credited with an affair and early marriage with a close friend of Hrithik Roshan, which in fact turned out to be very far from the truth.
Also see.

Bollywood superstar Hrithik Roshan turns 36 today. One of the most... talented actors Indian film industry, Hrithik captured the attention of the audience with his debut film "Kaho Na Pyaar Hai" in 2000.
Here is his life in photographs:

Hrithik was born in Bombay to a Punjabi family.


His father, actor and director Rakesh Roshan, is the son music director Roshana. His mother, Pinky, is the daughter of producer and director J Om Prakash. His uncle Rajesh Roshan is also a famous music director.


Hrithik started acting in films as a child in the 1980s. He made his debut as an actor in the film Aasha. He was only six years old.


Hrithik has remained in the limelight since 2000. His debut film Kaho Na Pyaar Hai became a blockbuster.




Hrithik played small roles in Aap Ke Deewane (1980) and Bhagwan Dada (1986), both of which his father Rakesh Roshan played the lead roles.


He then became an assistant director and worked in Karan Arjun (1995) and Koyla (1997).


Hrithik made his debut as a leading actor in the film "Kaho Naa Pyaar Hai" opposite Amisha Patel. He immediately became the heartthrob of the nation.


The film was a huge success, becoming the highest-grossing film of 2000. It also won the Filmfare Award for Best Film.


Hrithik won two Filmfare awards this year for Best Male Debut and Best Actor for Kaho Naa Pyaar Hai. The film entered the Limca Book of Records (Indian Book of Records) in 2003 for the most awards received by a Bollywood film (102 awards).


Hrithik married his childhood sweetheart Sussanne Khan, daughter of Khan Sanjay Khan, on December 20, 2000.

The wedding ceremony took place at the well-guarded Golden Palms Resort And Spa, on the Bangalore-Pune Highway.


It was a private ceremony to which only close friends and family of Hrithik and Sussanne were invited.


The couple has two sons, Hrehaan and Hridan.

Hrithik Roshan received huge recognition for his role in Khaled Mohammed's Fiza.


Although the film did not do well at the box office, it earned him a second Filmfare nomination for Best Actor.


Hrithik's Mission Kashmir became the third highest-grossing film of the year.


In 2001, Hrithik Roshan starred in Subhash Ghai's Yaadein.


Hrithik then starred in his first Karan Johar film, Kabhi Khushi Kabhie Gham, which did very well, becoming the second highest-grossing film of 2001 and the biggest hit abroad.


2002 was a bad year for Hrithik, with all three of his releases - Mujhse Dosti Karoge!, Na Tum Jaano Na Hum and Aapa Mujhe Achche Lagne Lage - managing to make a big impact at the box office.


In 2003, his luck returned to him in the science fiction film "Koi Mil Gaya". In the film, directed by Rakesh Roshan, Hrithik plays a mentally challenged young man.


Farhan Akhtar's 2004 film Lakshya did not do well at the box office. However, Hrithik's role was praised by critics.


Hrithik did two year break filming ahead of the superhero film Krrish, a sequel to his 2003 film Koi... Mil Gaya, which was released in June 2006.


The film was a huge box office success and became one of the best films of 2006.


For Dhoom 2, the sequel to 2004's Dhoom, he received his third Filmfare Award for Best Actor. The film subsequently became the highest-grossing film of 2006.


He looks like greek god in the film, for which he received rave reviews.


In 2008, Hrithik starred in Ashutosh Gowariker's Jodhaa Akbar, in which Aishwarya Rai-Bachchan became his partner.


He played the role of Akbar in the film. The film was very well received both in India and abroad.


"Jodhaa Akbar" brought him the first international award for Best Actor at the Golden Minbar International Film Festival in Kazan, Russia.


Hrithik recently made a special appearance in Zoya Akhtar's Luck by Chance.


Hrithik Roshan and Aishwarya Rai Bachchan are back for the third time with Sanjay Leela Bhansali's Guzaarish.


In 2009 family life Hrithika was put to the test when he and his Kites co-star Barbara Mori were spotted together at public events.


There were rumors that Hrithik's wife Sussanne was not happy with the growing closeness between Hrithik and Barbara. It was even said that Suzanne had left Hrithik's house.

Barbara plays his love interest in the upcoming film Kites. The film also stars Kangana Ranaut.

Gorgeous Hrithik Roshan (India)

Name: Hrithik Nagrath Roshan
The name was chosen by the grandmother, translated from Bengali as “saint”
Nationality: Indian
Date of birth: January 10, 1974 (Zodiac sign: Capricorn, Eastern calendar: Ox)
Nickname: Duggu Another home nickname is Bolunath.
Height:6’1”(inches.1inch=approx. 3cm)
Weight74kg.
Parents: Father Rakesh Roshan, mother Pinky Roshan
Sister: Sunaina, the eldest of the Roshan children, divorced, has a daughter, Suranika, born around 1997-98.
Wife: Suzanne Khan - daughter of director Sanjay Khan and Zarina Khan. Sister of Zayed Khan. Niece of Feroz Khan, cousin of Fardeen Khan.

Wedding: Hrithik and Suzanne took place on December 20, 2000
Children: two sons
Education: Sydney College
Favorite actresses: Madhubala, Madhuri, Julia Roberts
Favorite actors: Amitabh Bachchan, Raj Kapoor, Richard Gere, Al Pacino
Favorite artists: Bryan Adams, Britney Spears
Favorite color: Black
Pets: Persian cat named Tiger

Little details:

Hrithik Roshan's very first director was his maternal grandfather, J. Om Prakash, in Rakesh Roshan's Bhagwan Dada (1986), in which he acted while still a child.

On screen, Hrithik is always called by a name starting with “R”, which means “Romance”.
Except for Main Prem Ki Deewani Hoon where he was one of the two Prems.
However, in two action films, Fiza and Mission Kashmir, he was named with the letter “A”, that is, “Action”.

Kareena Kapoor is his most frequent partner,
and Udit Narayan is his most frequent voice-over singer.
Jaya Bachchan played his mother on screen twice.

Hrithik was born into an acting family, so he always knew that he would definitely be involved in cinema.
As a child, he had the ability to find unusual traits in people. He could imitate them very accurately. He was capable of playing even an alcoholic. And he was always excited about dancing. Be it a birthday or any other family occasion, he was on the dance floor until the very end.
Hrithik first appeared in films at the age of six in the film Aasha in 1980.
He also appeared in small roles in the films Aap Ki Deewane (1980) and Bhagwan Dada (1986).
Modest and shy, it was difficult to imagine him as a “movie hero,” and this played a role in his decision to study filmmaking abroad.
Returning to India, he became his father's third assistant director on the films Karan Arjun (1995) and Koyla (1997).

Hrithik led a grueling lifestyle, like any junior on location.
Dreaming of becoming an actor, he took Kishore Namit Kapoor's acting classes.
The next step into the “film future” was his first the main role. In 2000, he made his debut as a "hero" in the film Kaho Naa Pyaar Hai along with another debutant, actress Ameesha Patel. The film was directed by his father and proved to be very successful at the box office and also won the Filmfare Best Film of the Year. Hrithik's performance earned him the Filmfare Best Debutant and Filmfare Best Actor awards. The film Kaho Naa Pyaar Hai was included in the Limca Book of Records in 2003 as the film that won the most awards.
The instant “Ritiko-mania” that emerged gripped fans of Indian cinema all over the world.
In 2000, Hrithik starred in the films Fiza, which was not a commercial success, and Mission Kashmir, which was not a box office success. His next film Yaadein (2001) was also a flop. Then came Kabhi Khushi Kabhie Gham, which became the second highest grossing film of 2001.
2002 was a bad year for Hrithik. All three of his films - Mujhse Dosti Karoge!, Na Tum Jaano Na Hum and Aap Mujhe Achche Lagne Lage - flopped.
In 2003, his film Koi Mil Gaya was the most popular film of the year and won many Filmfare Awards, including Best Actor of the Year for Hrithik.
In 2004, only one of his films failed - Lakshya, although critics called it his most impressive work.
The actor did not act in film for two years. And in 2006 it was released New film Krrish. It was as successful as the 2003 film Koi Mil Gaya. Krrish was the biggest success of the year and one of the most box office films 2006.
His new film, Dhoom2 (2006), a sequel to the 2004 hit Dhoom, in which he played a negative role for the first time, also became one of the highest-grossing films of the year.
From heroes to failed heroes and then to superheroes. Such is the Bollywood life of Hrithik Roshan.

Hrithik Roshan is definitely a Bollywood icon. Millions of young people in India dream of having the same muscles and dancing skills, and millions of girls are ready to admire him for hours beautiful face. From the outside it may seem that he had every opportunity to become a superstar: dad - famous director, unparalleled appearance, outstanding not only as an actor, but also dance talent(something that very few people in Indian cinema can boast of).

Still from the film "Jodha and Akbar"

Among all other awards, Roshan won six Filmfare Awards, four of which were for Best Actor.

January 20, 2011 London Museum Madame Tussaud's wax figures have been replenished wax figure real size. Roshan came fifth Indian actor, whose wax statues are displayed in the museum.