The pyramid is correct. Pyramid

Pyramid. Truncated pyramid

Pyramid is a polyhedron, one of whose faces is a polygon ( base ), and all other faces are triangles with a common vertex ( side faces ) (Fig. 15). The pyramid is called correct , if its base is a regular polygon and the top of the pyramid is projected into the center of the base (Fig. 16). A triangular pyramid with all edges equal is called tetrahedron .

Lateral rib of a pyramid is the side of the side face that does not belong to the base Height pyramid is the distance from its top to the plane of the base. All lateral edges of a regular pyramid are equal to each other, all lateral faces are equal isosceles triangles. The height of the side face of a regular pyramid drawn from the vertex is called apothem . Diagonal section is called a section of a pyramid by a plane passing through two lateral edges that do not belong to the same face.

Lateral surface area pyramid is the sum of the areas of all lateral faces. Total surface area is called the sum of the areas of all the side faces and the base.

Theorems

1. If in a pyramid all the lateral edges are equally inclined to the plane of the base, then the top of the pyramid is projected into the center of the circle circumscribed near the base.

2. If all the side edges of a pyramid have equal lengths, then the top of the pyramid is projected into the center of a circle circumscribed near the base.

3. If all the faces in a pyramid are equally inclined to the plane of the base, then the top of the pyramid is projected into the center of a circle inscribed in the base.

To calculate the volume of an arbitrary pyramid, the correct formula is:

Where V- volume;

S base– base area;

H– height of the pyramid.

For a regular pyramid, the following formulas are correct:

Where p– base perimeter;

h a– apothem;

H- height;

S full

S side

S base– base area;

V– volume of a regular pyramid.

Truncated pyramid called the part of the pyramid enclosed between the base and a cutting plane parallel to the base of the pyramid (Fig. 17). Regular truncated pyramid called the part of a regular pyramid enclosed between the base and a cutting plane parallel to the base of the pyramid.

Reasons truncated pyramid - similar polygons. Side faces – trapezoids. Height of a truncated pyramid is the distance between its bases. Diagonal a truncated pyramid is a segment connecting its vertices that do not lie on the same face. Diagonal section is a section of a truncated pyramid by a plane passing through two lateral edges that do not belong to the same face.

For a truncated pyramid the following formulas are valid:

(4)

Where S 1 , S 2 – areas of the upper and lower bases;

S full– total surface area;

S side– lateral surface area;

H- height;

V– volume of a truncated pyramid.

For a regular truncated pyramid the formula is correct:

Where p 1 , p 2 – perimeters of the bases;

h a– apothem of a regular truncated pyramid.

Example 1. In a regular triangular pyramid, the dihedral angle at the base is 60º. Find the tangent of the angle of inclination of the side edge to the plane of the base.

Solution. Let's make a drawing (Fig. 18).

The pyramid is regular, which means that at the base there is an equilateral triangle and all the side faces are equal isosceles triangles. The dihedral angle at the base is the angle of inclination of the side face of the pyramid to the plane of the base. The linear angle is the angle a between two perpendiculars: etc. The top of the pyramid is projected at the center of the triangle (the center of the circumcircle and inscribed circle of the triangle ABC). The angle of inclination of the side edge (for example S.B.) is the angle between the edge itself and its projection onto the plane of the base. For the rib S.B. this angle will be the angle SBD. To find the tangent you need to know the legs SO And O.B.. Let the length of the segment BD equals 3 A. Dot ABOUT line segment BD is divided into parts: and From we find SO: From we find:

Answer:

Example 2. Find the volume of a regular truncated quadrangular pyramid if the diagonals of its bases are equal to cm and cm, and its height is 4 cm.

Solution. To find the volume of a truncated pyramid, we use formula (4). To find the area of ​​the bases, you need to find the sides of the base squares, knowing their diagonals. The sides of the bases are equal to 2 cm and 8 cm, respectively. This means the areas of the bases and Substituting all the data into the formula, we calculate the volume of the truncated pyramid:

Answer: 112 cm 3.

Example 3. Find the area of ​​the lateral face of a regular triangular truncated pyramid, the sides of the bases of which are 10 cm and 4 cm, and the height of the pyramid is 2 cm.

Solution. Let's make a drawing (Fig. 19).

The side face of this pyramid is an isosceles trapezoid. To calculate the area of ​​a trapezoid, you need to know the base and height. The bases are given according to the condition, only the height remains unknown. We'll find her from where A 1 E perpendicular from a point A 1 on the plane of the lower base, A 1 D– perpendicular from A 1 per AC. A 1 E= 2 cm, since this is the height of the pyramid. To find DE Let's make an additional drawing showing the top view (Fig. 20). Dot ABOUT– projection of the centers of the upper and lower bases. since (see Fig. 20) and On the other hand OK– radius inscribed in the circle and OM– radius inscribed in a circle:

MK = DE.

According to the Pythagorean theorem from

Side face area:

Answer:

Example 4. At the base of the pyramid lies an isosceles trapezoid, the bases of which A And b (a> b). Each side face forms an angle equal to the plane of the base of the pyramid j. Find the total surface area of ​​the pyramid.

Solution. Let's make a drawing (Fig. 21). Total surface area of ​​the pyramid SABCD equal to the sum of the areas and the area of ​​the trapezoid ABCD.

Let us use the statement that if all the faces of the pyramid are equally inclined to the plane of the base, then the vertex is projected into the center of the circle inscribed in the base. Dot ABOUT– vertex projection S at the base of the pyramid. Triangle SOD is the orthogonal projection of the triangle CSD to the plane of the base. Using the theorem on the area of ​​the orthogonal projection of a plane figure, we obtain:

Likewise it means Thus, the problem was reduced to finding the area of ​​the trapezoid ABCD. Let's draw a trapezoid ABCD separately (Fig. 22). Dot ABOUT– the center of a circle inscribed in a trapezoid.

Since a circle can be inscribed in a trapezoid, then or From the Pythagorean theorem we have

Introduction

When we started studying stereometric figures, we touched on the topic “Pyramid”. We liked this topic because the pyramid is very often used in architecture. And since our future profession of architecture is inspired by this figure, we think that she can push us towards excellent projects.

The strength of architectural structures is their most important quality. Linking strength, firstly, with the materials from which they are created, and, secondly, with the features of design solutions, it turns out that the strength of a structure is directly related to the geometric shape that is basic for it.

In other words, we are talking about a geometric figure that can be considered as a model of the corresponding architectural form. It turns out that geometric shape also determines the strength of an architectural structure.

Since ancient times, the Egyptian pyramids have been considered the most durable architectural structures. As you know, they have the shape of regular quadrangular pyramids.

It is this geometric shape that provides the greatest stability due to the large base area. On the other hand, the pyramid shape ensures that the mass decreases as the height above the ground increases. It is these two properties that make the pyramid stable, and therefore strong under the conditions of gravity.

Objective of the project: learn something new about pyramids, deepen your knowledge and find practical application.

To achieve this goal, it was necessary to solve the following tasks:

· Learn historical information about the pyramid

· Consider the pyramid as a geometric figure

· Find application in life and architecture

· Find similarities and differences between pyramids located in different parts of the world

Theoretical part

Historical information

Pyramid geometry began in Ancient Egypt and Babylon, but was actively developed in Ancient Greece. The first to establish the volume of the pyramid was Democritus, and Eudoxus of Cnidus proved it. The ancient Greek mathematician Euclid systematized knowledge about the pyramid in the XII volume of his “Elements”, and also derived the first definition of a pyramid: a solid figure bounded by planes that converge from one plane to one point.

Tombs of Egyptian pharaohs. The largest of them - the pyramids of Cheops, Khafre and Mikerin in El Giza - were considered one of the Seven Wonders of the World in ancient times. The construction of the pyramid, in which the Greeks and Romans already saw a monument to the unprecedented pride of kings and cruelty that doomed the entire people of Egypt to meaningless construction, was the most important cult act and was supposed to express, apparently, the mystical identity of the country and its ruler. The population of the country worked on the construction of the tomb during the part of the year free from agricultural work. A number of texts testify to the attention and care that the kings themselves (albeit of a later time) paid to the construction of their tomb and its builders. It is also known about the special cult honors that were given to the pyramid itself.

Basic Concepts

Pyramid is called a polyhedron whose base is a polygon, and the remaining faces are triangles that have a common vertex.

Apothem- the height of the side face of a regular pyramid, drawn from its vertex;

Side faces- triangles meeting at a vertex;

Side ribs- common sides of the side faces;

Top of the pyramid- a point connecting the side ribs and not lying in the plane of the base;

Height- a perpendicular segment drawn through the top of the pyramid to the plane of its base (the ends of this segment are the top of the pyramid and the base of the perpendicular);

Diagonal section of a pyramid- section of the pyramid passing through the top and diagonal of the base;

Base- a polygon that does not belong to the vertex of the pyramid.

Basic properties of a regular pyramid

The lateral edges, lateral faces and apothems are respectively equal.

The dihedral angles at the base are equal.

The dihedral angles at the lateral edges are equal.

Each height point is equidistant from all the vertices of the base.

Each height point is equidistant from all side faces.

Basic pyramid formulas

The area of ​​the lateral and total surface of the pyramid.

The area of ​​the lateral surface of a pyramid (full and truncated) is the sum of the areas of all its lateral faces, the total surface area is the sum of the areas of all its faces.

Theorem: 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 of the pyramid.

p- base perimeter;

h- apothem.

The area of ​​the lateral and full surfaces of a truncated pyramid.

p 1, p 2 - base perimeters;

h- apothem.

R- total surface area of ​​a regular truncated pyramid;

S side- area of ​​the lateral surface of a regular truncated pyramid;

S 1 + S 2- base area

Volume of the pyramid

Form volume ula is used for pyramids of any kind.

H- height of the pyramid.

Pyramid corners

The angles formed by the side face and the base of the pyramid are called dihedral angles at the base of the pyramid.

A dihedral angle is formed by two perpendiculars.

To determine this angle, you often need to use the three perpendicular theorem.

The angles formed by the lateral edge and its projection onto the base plane are called angles between the side edge and the plane of the base.

The angle formed by two lateral edges is called dihedral angle at the lateral edge of the pyramid.

The angle formed by two lateral edges of one face of the pyramid is called angle at the top of the pyramid.

Pyramid sections

The surface of a pyramid is the surface of a polyhedron. Each of its faces is a plane, therefore the section of a pyramid defined by a cutting plane is a broken line consisting of individual straight lines.

Diagonal section

The section of a pyramid by a plane passing through two lateral edges that do not lie on the same face is called diagonal section pyramids.

Parallel sections

Theorem:

If the pyramid is intersected by a plane parallel to the base, then the lateral edges and heights of the pyramid are divided by this plane into proportional parts;

The section of this plane is a polygon similar to the base;

The areas of the section and the base are related to each other as the squares of their distances from the vertex.

Types of pyramid

Correct pyramid– a pyramid whose base is a regular polygon, and the top of the pyramid is projected into the center of the base.

For a regular pyramid:

1. side ribs are equal

2. side faces are equal

3. apothems are equal

4. dihedral angles at the base are equal

5. dihedral angles at the lateral edges are equal

6. each point of height is equidistant from all vertices of the base

7. each height point is equidistant from all side edges

Truncated pyramid- part of the pyramid enclosed between its base and a cutting plane parallel to the base.

The base and corresponding section of a truncated pyramid are called bases of a truncated pyramid.

A perpendicular drawn from any point of one base to the plane of another is called the height of a truncated pyramid.

Tasks

No. 1. In a regular quadrangular pyramid, point O is the center of the base, SO=8 cm, BD=30 cm. Find the side edge SA.

Problem solving

No. 1. In a regular pyramid, all faces and edges are equal.

Consider OSB: OSB is a rectangular rectangle, because.

SB 2 =SO 2 +OB 2

SB 2 =64+225=289

Pyramid in architecture

A pyramid is a monumental structure in the form of an ordinary regular geometric pyramid, in which the sides converge at one point. According to their functional purpose, pyramids in ancient times were places of burial or cult worship. The base of a pyramid can be triangular, quadrangular, or in the shape of a polygon with an arbitrary number of vertices, but the most common version is the quadrangular base.

There are a considerable number of pyramids built by different cultures of the Ancient World, mainly as temples or monuments. Large pyramids include the Egyptian pyramids.

All over the Earth you can see architectural structures in the form of pyramids. The pyramid buildings are reminiscent of ancient times and look very beautiful.

Egyptian pyramids are the greatest architectural monuments of Ancient Egypt, including one of the “Seven Wonders of the World”, the Pyramid of Cheops. From the foot to the top it reaches 137.3 m, and before it lost the top, its height was 146.7 m

The radio station building in the capital of Slovakia, resembling an inverted pyramid, was built in 1983. In addition to offices and service premises, inside the volume there is a fairly spacious concert hall, which has one of the largest organs in Slovakia.

The Louvre, which is “silent, unchanged and majestic, like a pyramid,” has undergone many changes over the centuries before becoming the greatest museum in the world. It was born as a fortress, erected by Philip Augustus in 1190, which soon became a royal residence. In 1793 the palace became a museum. Collections are enriched through bequests or purchases.

Video tutorial 2: Pyramid problem. Volume of the pyramid

Video tutorial 3: Pyramid problem. Correct pyramid

Lecture: Pyramid, its base, lateral ribs, height, lateral surface; triangular pyramid; regular pyramid

Pyramid is a three-dimensional body that has a polygon at its base, and all its faces consist of triangles.

A special case of a pyramid is a cone with a circle at its base.

Let's look at the main elements of the pyramid:

Apothem- this is a segment that connects the top of the pyramid with the middle of the lower edge of the side face. In other words, this is the height of the edge of the pyramid.

In the figure you can see triangles ADS, ABS, BCS, CDS. If you look closely at the names, you can see that each triangle has one common letter in its name - S. That is, this means that all the side faces (triangles) converge at one point, which is called the top of the pyramid.

The segment OS that connects the vertex with the point of intersection of the diagonals of the base (in the case of triangles - at the point of intersection of the heights) is called pyramid height.

A diagonal section is a plane that passes through the top of the pyramid, as well as one of the diagonals of the base.

Since the side surface of the pyramid consists of triangles, to find the total area of ​​the side surface it is necessary to find the area of ​​each face and add them up. The number and shape of faces depends on the shape and size of the sides of the polygon that lies at the base.

The only plane in a pyramid that does not belong to its vertex is called basis pyramids.

In the figure we see that the base is a parallelogram, however, it can be any arbitrary polygon.

Properties:

Consider the first case of a pyramid, in which it has edges of the same length:

• A circle can be drawn around the base of such a pyramid. If you project the top of such a pyramid, then its projection will be located in the center of the circle.
• The angles at the base of the pyramid are the same on each face.
• In this case, a sufficient condition for the fact that a circle can be described around the base of the pyramid, and also that all the edges are of different lengths, can be considered the same angles between the base and each edge of the faces.

If you come across a pyramid in which the angles between the side faces and the base are equal, then the following properties are true:

• You will be able to describe a circle around the base of the pyramid, the apex of which is projected exactly at the center.
• If you draw each side edge of the height to the base, then they will be of equal length.
• To find the lateral surface area of ​​such a pyramid, it is enough to find the perimeter of the base and multiply it by half the length of the height.
• S bp = 0.5P oc H.
• Types of pyramid.
• Depending on which polygon lies at the base of the pyramid, they can be triangular, quadrangular, etc. If at the base of the pyramid there is a regular polygon (with equal sides), then such a pyramid will be called regular.

Regular triangular pyramid

Definition. Side edge- this is a triangle in which one angle lies at the top of the pyramid, and the opposite side coincides with the side of the base (polygon).

Definition. Side ribs- these are the common sides of the side faces. A pyramid has as many edges as the angles of a polygon.

Definition. Pyramid height- this is a perpendicular lowered from the top to the base of the pyramid.

Definition. Apothem- this is a perpendicular to the side face of the pyramid, lowered from the top of the pyramid to the side of the base.

Definition. Diagonal section- this is a section of a pyramid by a plane passing through the top of the pyramid and the diagonal of the base.

Definition. Correct pyramid is a pyramid in which the base is a regular polygon, and the height descends to the center of the base.

Volume and surface area of ​​the pyramid

Formula. Volume of the pyramid through base area and height:

Properties of the pyramid

If all the side edges are equal, then a circle can be drawn around the base of the pyramid, and the center of the base coincides with the center of the circle. Also, a perpendicular dropped from the top passes through the center of the base (circle).

If all the side edges are equal, then they are inclined to the plane of the base at the same angles.

The lateral edges are equal when they form equal angles with the plane of the base or if a circle can be described around the base of the pyramid.

If the side faces are inclined to the plane of the base at the same angle, then a circle can be inscribed into the base of the pyramid, and the top of the pyramid is projected into its center.

If the side faces are inclined to the plane of the base at the same angle, then the apothems of the side faces are equal.

Properties of a regular pyramid

1. The top of the pyramid is equidistant from all corners of the base.

2. All side edges are equal.

3. All side ribs are inclined at equal angles to the base.

4. The apothems of all lateral faces are equal.

5. The areas of all side faces are equal.

6. All faces have the same dihedral (flat) angles.

7. A sphere can be described around the pyramid. The center of the circumscribed sphere will be the intersection point of the perpendiculars that pass through the middle of the edges.

8. You can fit a sphere into a pyramid. The center of the inscribed sphere will be the point of intersection of the bisectors emanating from the angle between the edge and the base.

9. If the center of the inscribed sphere coincides with the center of the circumscribed sphere, then the sum of the plane angles at the vertex is equal to π or vice versa, one angle is equal to π/n, where n is the number of angles at the base of the pyramid.

The connection between the pyramid and the sphere

A sphere can be described around a pyramid when at the base of the pyramid there is a polyhedron around which a circle can be described (a necessary and sufficient condition). The center of the sphere will be the intersection point of planes passing perpendicularly through the midpoints of the side edges of the pyramid.

It is always possible to describe a sphere around any triangular or regular pyramid.

A sphere can be inscribed in a pyramid if 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.

Connection of a pyramid with a cone

A cone is said to be inscribed in a pyramid if their vertices coincide and the base of the cone is inscribed in the base of the pyramid.

A cone can be inscribed in a pyramid if the apothems of the pyramid are equal to each other.

A cone is said to be circumscribed around a pyramid if their vertices coincide and the base of the cone is circumscribed around the base of the pyramid.

A cone can be described around a pyramid if all the lateral edges of the pyramid are equal to each other.

Relationship between a pyramid and a cylinder

A pyramid is called inscribed in a cylinder if the top of the pyramid lies on one base of the cylinder, and the base of the pyramid is inscribed in another base of the cylinder.

A cylinder can be described around a pyramid if a circle can be described around the base of the pyramid.

A tetrahedron has four faces and four vertices and six edges, where any two edges do not have common vertices but do not touch.

Each vertex consists of three faces and edges that form triangular angle.

The segment connecting the vertex of a tetrahedron with the center of the opposite face is called median of the tetrahedron(GM).

Bimedian called a segment connecting the midpoints of opposite edges that do not touch (KL).

All bimedians and medians of a tetrahedron intersect at one point (S). In this case, the bimedians are divided in half, and the medians are divided in a ratio of 3:1 starting from the top.

Definition. Acute angled pyramid- a pyramid in which the apothem is more than half the length of the side of the base.

Definition. Obtuse pyramid- a pyramid in which the apothem is less than half the length of the side of the base.

Definition. Regular tetrahedron- a tetrahedron in which all four faces are equilateral triangles. It is one of the five regular polygons. In a regular tetrahedron, all dihedral angles (between faces) and trihedral angles (at the vertex) are equal.

Definition. Rectangular tetrahedron is called a tetrahedron in which there is a right angle between three edges at the apex (the edges are perpendicular). Three faces form rectangular triangular angle and the faces are right triangles, and the base is an arbitrary triangle. The apothem of any face is equal to half the side of the base on which the apothem falls.

Definition. Isohedral tetrahedron is called a tetrahedron whose side faces are equal to each other, and the base is a regular triangle. Such a tetrahedron has faces that are isosceles triangles.

Definition. Orthocentric tetrahedron is called a tetrahedron in which all the heights (perpendiculars) that are lowered from the top to the opposite face intersect at one point.

Definition. Star pyramid called a polyhedron whose base is a star.