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The word “pyramid” is involuntarily associated with the majestic giants in Egypt, faithfully guarding the peace of the pharaohs. Maybe that’s why everyone, even children, recognizes the pyramid unmistakably.

Nevertheless, let's try to give it a geometric definition. Let us imagine several points on the plane (A1, A2,..., An) and one more (E) that does not belong to it. So, if point E (vertex) is connected to the vertices of the polygon formed by points A1, A2,..., An (base), you get a polyhedron, which is called a pyramid. Obviously, the polygon at the base of the pyramid can have any number of vertices, and depending on their number, the pyramid can be called triangular, quadrangular, pentagonal, etc.

If you look closely at the pyramid, it will become clear why it is also defined in another way - as a geometric figure with a polygon at its base, and triangles united by a common vertex as its side faces.

Since a pyramid is a spatial figure, it also has one quantitative characteristic, as calculated from the well-known equal third of the product of the base of the pyramid and its height:

When deriving the formula, the volume of a pyramid is initially calculated for a triangular one, taking as a basis a constant ratio connecting this value with the volume of a triangular prism having the same base and height, which, as it turns out, is three times this volume.

And since any pyramid is divided into triangular ones, and its volume does not depend on the constructions performed during the proof, the validity of the given volume formula is obvious.

Standing apart among all the pyramids are the correct ones, which have at their base As for, it should “end” in the center of the base.

In the case of an irregular polygon at the base, to calculate the area of ​​the base you will need:

• break it into triangles and squares;

• calculate the area of ​​each of them;

• add up the received data.

In the case of a regular polygon at the base of the pyramid, its area is calculated using ready-made formulas, so the volume regular pyramid It is calculated quite simply.

For example, to calculate the volume of a quadrangular pyramid, if it is regular, the length of the side of a regular quadrilateral (square) at the base is squared and, multiplied by the height of the pyramid, the resulting product is divided by three.

The volume of the pyramid can be calculated using other parameters:

• as a third of the product of the radius of a ball inscribed in a pyramid and its total surface area;

• as two-thirds of the product of the distance between two arbitrarily chosen crossing edges and the area of ​​the parallelogram that forms the midpoints of the remaining four edges.

The volume of a pyramid is calculated simply in the case when its height coincides with one of the side edges, that is, in the case of a rectangular pyramid.

Speaking about pyramids, we cannot ignore truncated pyramids, obtained by cutting the pyramid with a plane parallel to the base. Their volume is almost equal to the difference between the volumes of the whole pyramid and the cut off top.

The first is the volume of the pyramid, although not entirely in its modern form, however, equal to 1/3 of the volume of the prism known to us, Democritus found. Archimedes called his method of calculation “without proof,” since Democritus approached the pyramid as a figure composed of infinitely thin, similar plates.

Vector algebra also “addressed” the issue of finding the volume of a pyramid, using the coordinates of its vertices. Pyramid built on three vectors a,b,c, is equal to one sixth of the modulus of the mixed product of the given vectors.

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 ribs are equal when they form with the plane of the base equal angles 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 A polyhedron whose base is a star is called.

One of the simplest volumetric figures is a triangular pyramid, since it consists of the smallest number of faces from which a figure can be formed in space. In this article we will look at formulas that can be used to find the volume of a triangular regular pyramid.

Triangular pyramid

According to general definition a pyramid is a polygon, all of whose vertices are connected to one point not located in the plane of this polygon. If the latter is a triangle, then the entire figure is called a triangular pyramid.

The pyramid in question consists of a base (triangle) and three side faces (triangles). The point at which the three side faces are connected is called the vertex of the figure. The perpendicular from this vertex dropped to the base is the height of the pyramid. If the point of intersection of the perpendicular with the base coincides with the point of intersection of the medians of the triangle at the base, then we speak of a regular pyramid. Otherwise it will be slanted.

As stated, the base of a triangular pyramid can be a general type of triangle. However, if it is equilateral, and the pyramid itself is straight, then they speak of a regular three-dimensional figure.

Any one has 4 faces, 6 edges and 4 vertices. If the lengths of all edges are equal, then such a figure is called a tetrahedron.

general type

Before writing down a regular triangular pyramid, we give an expression for this physical quantity for a general type pyramid. This expression looks like:

Here S o is the area of ​​the base, h is the height of the figure. This equality will be valid for any type of pyramid polygon base, as well as for a cone. If at the base there is a triangle with side length a and height h o lowered onto it, then the formula for volume will be written as follows:

Formulas for the volume of a regular triangular pyramid

Triangular has an equilateral triangle at the base. It is known that the height of this triangle is related to the length of its side by the equality:

Substituting this expression into the formula for the volume of a triangular pyramid written in the previous paragraph, we obtain:

V = 1/6*a*h o *h = √3/12*a 2 *h.

The volume of a regular pyramid with a triangular base is a function of the length of the side of the base and the height of the figure.

Since any regular polygon can be inscribed in a circle, the radius of which will uniquely determine the length of the side of the polygon, then this formula can be written in terms of the corresponding radius r:

This formula can be easily obtained from the previous one, if we take into account that the radius r of the circumscribed circle through the length of side a of the triangle is determined by the expression:

Problem of determining the volume of a tetrahedron

We will show how to use the above formulas when solving specific geometry problems.

It is known that a tetrahedron has an edge length of 7 cm. Find the volume of a regular triangular pyramid-tetrahedron.

Recall that a tetrahedron is a regular triangular pyramid in which all bases are equal to each other. To use the formula for the volume of a regular triangular pyramid, you need to calculate two quantities:

• length of the side of the triangle;
• height of the figure.

The first quantity is known from the problem statement:

To determine the height, consider the figure shown in the figure.

The marked triangle ABC is a right triangle, where angle ABC is 90 o. Side AC is the hypotenuse and its length is a. Using simple geometric reasoning, it can be shown that side BC has the length:

Note that the length BC is the radius of the circle circumscribed around the triangle.

h = AB = √(AC 2 - BC 2) = √(a 2 - a 2 /3) = a*√(2/3).

Now you can substitute h and a into the corresponding formula for volume:

V = √3/12*a 2 *a*√(2/3) = √2/12*a 3 .

Thus, we have obtained the formula for the volume of a tetrahedron. It can be seen that the volume depends only on the length of the edge. If we substitute the value from the problem condition into the expression, then we get the answer:

V = √2/12*7 3 ≈ 40.42 cm 3.

If we compare this value with the volume of a cube having the same edge, we find that the volume of the tetrahedron is 8.5 times less. This indicates that the tetrahedron is a compact figure that occurs in some natural substances. For example, the methane molecule has a tetrahedral shape, and each carbon atom in diamond is connected to four other atoms to form a tetrahedron.

Homothetic pyramid problem

Let's solve one interesting geometric problem. Suppose that there is a triangular regular pyramid with a certain volume V 1. How many times should the size of this figure be reduced in order to obtain a homothetic pyramid with a volume three times smaller than the original?

Let's start solving the problem by writing the formula for the original regular pyramid:

V 1 = √3/12*a 1 2 *h 1 .

Let the volume of the figure required by the conditions of the problem be obtained by multiplying its parameters by the coefficient k. We have:

V 2 = √3/12*k 2 *a 1 2 *k*h 1 = k 3 *V 1 .

Since the ratio of the volumes of the figures is known from the condition, we obtain the value of the coefficient k:

k = ∛(V 2 /V 1) = ∛(1/3) ≈ 0.693.

Note that we would obtain a similar value for the coefficient k for a pyramid of any type, and not just for a regular triangular one.

Theorem.

The volume of the pyramid is equal to one third of the product of the area of ​​the base and the height.

Proof:

1. Consider a triangular pyramidOABCwith volume V, base areaS and height h. Let's draw the axis oh (OM2- height), consider the sectionA1 B1 C1pyramid with a plane perpendicular to the axisOhand, therefore, parallel to the plane of the base. Let us denote byX abscissa point M1 intersection of this plane with the x axis, and throughS(x)- cross-sectional area. Let's express S(x) through S, h And X. Note that triangles A1 IN1 WITH1 And ABCs are similar. Indeed A1 IN1 II AB, so triangle OA 1 IN 1 similar to triangle OAB. WITH therefore, A1 IN1 : AB= OA 1: OA .

Right Triangles OA 1 IN 1 and OAV are also similar (they have a common acute angle with vertex O). Therefore, OA 1: OA = O 1 M1 : OM = x: h. Thus A 1 IN 1 : A B = x: h.Similarly, it is proved thatB1 C1:Sun = X: h And A1 C1:AC = X: h.So, triangleA1 B1 C1 And ABCsimilar with similarity coefficient X: h.Therefore, S(x) : S = (x: h)², or S(x) = S x²/ h².

Let us now apply the basic formula for calculating the volumes of bodies ata= 0, b =h we get

Thus, the volume of the original pyramid is 1/3Sh. The theorem has been proven.

Consequence:

Volume V of a truncated pyramid whose height is h and whose base areas are S and S1 , are calculated by the formula

h - height of the pyramid

S top

- area of ​​the upper base

S lower - area of ​​the lower base The main characteristic of any

geometric figure

in space is its volume. In this article we will look at what a pyramid with a triangle at the base is, and we will also show how to find the volume of a triangular pyramid - regular full and truncated. What is this - a triangular pyramid? Everyone has heard of the ancients

Let's take an arbitrary triangle and connect all its vertices with some single point located outside the plane of this triangle. The resulting figure will be called a triangular pyramid. It is shown in the figure below.

As you can see, the figure in question is formed by four triangles, which in the general case are different. Each triangle is the sides of the pyramid or its face. This pyramid is often called a tetrahedron, that is, a tetrahedral three-dimensional figure.

In addition to the sides, the pyramid also has edges (there are 6 of them) and vertices (of 4).

with triangular base

A figure that is obtained using arbitrary triangle and points in space, will be an irregular inclined pyramid in the general case. Now imagine that the original triangle has identical sides, and a point in space is located exactly above its geometric center at a distance h from the plane of the triangle. The pyramid constructed using these initial data will be correct.

Obviously, the number of edges, sides and vertices of a regular triangular pyramid will be the same as that of a pyramid built from an arbitrary triangle.

However, the correct figure has some distinctive features:

• its height drawn from the top will exactly intersect the base at geometric center(point of intersection of medians);
• the lateral surface of such a pyramid is formed by three identical triangles, which are isosceles or equilateral.

A regular triangular pyramid is not only a purely theoretical geometric object. Some structures in nature have its shape, for example crystal cell a diamond, where a carbon atom is connected to four of the same atoms by covalent bonds, or a methane molecule, where the tops of the pyramid are formed by hydrogen atoms.

triangular pyramid

You can determine the volume of absolutely any pyramid with an arbitrary n-gon at the base using the following expression:

Here the symbol S o denotes the area of ​​the base, h is the height of the figure drawn to the marked base from the top of the pyramid.

Since the area of ​​an arbitrary triangle is equal to half the product of the length of its side a and the apothem h a dropped onto this side, the formula for the volume of a triangular pyramid can be written as follows:

V = 1/6 × a × h a × h

For the general type, determining the height is not an easy task. To solve it, the easiest way is to use the formula for the distance between a point (vertex) and a plane (triangular base), represented by the equation general view.

For the correct one, it has a specific appearance. The area of ​​the base (of an equilateral triangle) for it is equal to:

Substituting it into the general expression for V, we get:

V = √3/12 × a 2 × h

A special case is the situation when all sides of a tetrahedron turn out to be identical equilateral triangles. In this case, its volume can be determined only based on knowledge of the parameter of its edge a. The corresponding expression looks like:

Truncated pyramid

If top part, containing the vertex, cut off from a regular triangular pyramid, you get a truncated figure. Unlike the original one, it will consist of two equilateral triangular bases and three isosceles trapezoids.

The photo below shows what a regular truncated triangular pyramid made of paper looks like.

To determine the volume of a truncated triangular pyramid, you need to know its three linear characteristics: each of the sides of the bases and the height of the figure, equal to the distance between the upper and lower bases. The corresponding formula for volume is written as follows:

V = √3/12 × h × (A 2 + a 2 + A × a)

Here h is the height of the figure, A and a are the lengths of the large (lower) and small (upper) sides equilateral triangles respectively.

The solution of the problem

To make the information in the article clearer for the reader, we will show clear example, how to use some of the written formulas.

Let the volume of the triangular pyramid be 15 cm 3. It is known that the figure is correct. It is necessary to find the apothem a b of the lateral edge if it is known that the height of the pyramid is 4 cm.

Since the volume and height of the figure are known, you can use the appropriate formula to calculate the length of the side of its base. We have:

V = √3/12 × a 2 × h =>

a = 12 × V / (√3 × h) = 12 × 15 / (√3 × 4) = 25.98 cm

a b = √(h 2 + a 2 / 12) = √(16 + 25.98 2 / 12) = 8.5 cm

The calculated length of the apothem of the figure turned out to be greater than its height, which is true for any type of pyramid.