How many side faces does a triangular pyramid have? Pyramid. Formulas and properties of the pyramid

16.10.2019

Here you can find basic information about pyramids and related formulas and concepts. All of them are studied with a mathematics tutor in preparation for the Unified State Exam.

Consider a plane, a polygon , lying in it and a point S, not lying in it. Let's connect S to all the vertices of the polygon. The resulting polyhedron is called a pyramid. The segments are called side ribs. The polygon is called the base, and point S is the top of the pyramid. Depending on the number n, the pyramid is called triangular (n=3), quadrangular (n=4), pentagonal (n=5) and so on. An alternative name for a triangular pyramid is tetrahedron. The height of a pyramid is the perpendicular descending from its top to the plane of the base.

A pyramid is called regular if a regular polygon, and the base of the pyramid's altitude (the base of the perpendicular) is its center.

Tutor's comment:
Do not confuse the concepts of “regular pyramid” and “regular tetrahedron”. U regular pyramid the side edges are not necessarily equal to the edges of the base, but in a regular tetrahedron all 6 edges are equal. This is his definition. It is easy to prove that the equality implies the coincidence of the center P of the polygon with a base height, so a regular tetrahedron is a regular pyramid.

What is an apothem?
The apothem of a pyramid is the height of its side face. If the pyramid is regular, then all its apothems are equal. The reverse is not true.

A mathematics tutor about his terminology: 80% of work with pyramids is built through two types of triangles:
1) Containing apothem SK and height SP
2) Containing the lateral edge SA and its projection PA

To simplify references to these triangles, it is more convenient for a math tutor to call the first of them apothemic, and the second costal. Unfortunately, you will not find this terminology in any of the textbooks, and the teacher has to introduce it unilaterally.

Formula for the volume of a pyramid:
1) , where is the area of the base of the pyramid, and is the height of the pyramid
2) , where is the radius of the inscribed sphere, and is the area full surface pyramids.
3) , where MN is the distance between any two crossing edges, and is the area of the parallelogram formed by the midpoints of the four remaining edges.

Property of the base of the height of a pyramid:

Point P (see figure) coincides with the center of the inscribed circle at the base of the pyramid if one of the following conditions is met:
1) All apothems are equal
2) All side faces are equally inclined to the base
3) All apothems are equally inclined to the height of the pyramid
4) The height of the pyramid is equally inclined to all side faces

Math tutor's comment: Please note that all points are united by one common property: one way or another, lateral faces are involved everywhere (apothems are their elements). Therefore, the tutor can offer a less accurate, but more convenient for learning, formulation: point P coincides with the center of the inscribed circle, the base of the pyramid, if there is any equal information about its lateral faces. To prove it, it is enough to show that all apothem triangles are equal.

Point P coincides with the center of a circle circumscribed near the base of the pyramid if one of three conditions is true:
1) All side edges are equal
2) All side ribs are equally inclined to the base
3) All side ribs are equally inclined to the height

apothem- the height of the side face of a regular pyramid, which is drawn from its vertex (in addition, the apothem is the length of the perpendicular, which is lowered from the middle of the regular polygon to one of its sides);
side faces (ASB, BSC, CSD, DSA) - triangles that meet at the vertex;
lateral ribs ( AS , B.S. , C.S. , D.S. ) — common sides of the side faces;
top of the pyramid (t. S) - a point that connects the side ribs and which does not lie in the plane of the base;
height ( SO ) - a perpendicular segment drawn through the top of the pyramid to the plane of its base (the ends of such a segment will be the top of the pyramid and the base of the perpendicular);
diagonal section of the pyramid- a section of the pyramid that passes through the top and the diagonal of the base;
base (ABCD) - a polygon that does not belong to the vertex of the pyramid.

Properties of the pyramid.

1. When all side edges are of the same size, then:

it is easy to describe a circle near the base of the pyramid, and the top of the pyramid will be projected into the center of this circle;
the side ribs form equal angles with the plane of the base;
Moreover, the opposite is also true, i.e. when the lateral ribs form with the plane of the base equal angles, or when a circle can be described near the base of the pyramid and the top of the pyramid will be projected into the center of this circle, which means that all the side edges of the pyramid are the same size.

2. When the side faces have an angle of inclination to the plane of the base of the same value, then:

it is easy to describe a circle near the base of the pyramid, and the top of the pyramid will be projected into the center of this circle;
the heights of the side faces are of equal length;
the area of the side surface is equal to ½ the product of the perimeter of the base and the height of the side face.

3. A sphere can be described around a pyramid if at the base of the pyramid there is a polygon around which a circle can be described (a necessary and sufficient condition). The center of the sphere will be the point of intersection of the planes that pass through the middles of the edges of the pyramid perpendicular to them. From this theorem we conclude that a sphere can be described both around any triangular and around any regular pyramid.

4. A sphere can be inscribed into a pyramid if the bisector planes of the internal dihedral angles of the pyramid intersect at the 1st point (a necessary and sufficient condition). This point will become the center of the sphere.

The simplest pyramid.

Based on the number of angles, the base of the pyramid is divided into triangular, quadrangular, and so on.

There will be a pyramid triangular, quadrangular, and so on, when the base of the pyramid is a triangle, a quadrangle, and so on. A triangular pyramid is a tetrahedron - a tetrahedron. Quadrangular - pentagonal and so on.

Pyramid concept

Definition 1

Geometric figure, formed by a polygon and a point not lying in the plane containing this polygon, connected to all the vertices of the polygon is called a pyramid (Fig. 1).

The polygon from which the pyramid is made is called the base of the pyramid; the resulting triangles, when connected to a point, are the side faces of the pyramid, the sides of the triangles are the sides of the pyramid, and the point common to all triangles is the top of the pyramid.

Types of pyramids

Depending on the number of angles at the base of the pyramid, it can be called triangular, quadrangular, and so on (Fig. 2).

Figure 2.

Another type of pyramid is the regular pyramid.

Let us introduce and prove the property of a regular pyramid.

Theorem 1

All lateral faces of a regular pyramid are isosceles triangles that are equal to each other.

Proof.

Consider a regular $n-$gonal pyramid with vertex $S$ of height $h=SO$. Let us draw a circle around the base (Fig. 4).

Figure 4.

Consider the triangle $SOA$. According to the Pythagorean theorem, we get

Obviously, any side edge will be defined this way. Consequently, all side edges are equal to each other, that is, all side faces are isosceles triangles. Let us prove that they are equal to each other. Since the base is a regular polygon, the bases of all side faces are equal to each other. Consequently, all lateral faces are equal according to the III criterion of equality of triangles.

The theorem has been proven.

Let us now introduce the following definition related to the concept of a regular pyramid.

Definition 3

The apothem of a regular pyramid is the height of its side face.

Obviously, by Theorem One, all apothems are equal to each other.

Theorem 2

The lateral surface area of a regular pyramid is determined as the product of the semi-perimeter of the base and the apothem.

Proof.

Let us denote the side of the base of the $n-$gonal pyramid by $a$, and the apothem by $d$. Therefore, the area of the side face is equal to

Since, according to Theorem 1, everything sides are equal, then

The theorem has been proven.

Another type of pyramid is a truncated pyramid.

Definition 4

If a plane parallel to its base is drawn through an ordinary pyramid, then the figure formed between this plane and the plane of the base is called a truncated pyramid (Fig. 5).

Figure 5. Truncated pyramid

The lateral faces of the truncated pyramid are trapezoids.

Theorem 3

The lateral surface area of a regular truncated pyramid is determined as the product of the sum of the semi-perimeters of the bases and the apothem.

Proof.

Let us denote the sides of the bases of the $n-$gonal pyramid by $a\ and\ b$, respectively, and the apothem by $d$. Therefore, the area of the side face is equal to

Since all sides are equal, then

The theorem has been proven.

Sample task

Example 1

Find the area of the lateral surface of a truncated triangular pyramid if it is obtained from a regular pyramid with base side 4 and apothem 5 by cutting off a plane passing through the midline of the side faces.

Solution.

By the theorem about midline we find that the upper base of the truncated pyramid is equal to $4\cdot \frac(1)(2)=2$, and the apothem is equal to $5\cdot \frac(1)(2)=2.5$.

Then, by Theorem 3, we get

Definition

Pyramid is a polyhedron composed of a polygon $A_1A_2...A_n$ and $n$ triangles with a common vertex $P$ (not lying in the plane of the polygon) and sides opposite it, coinciding with the sides of the polygon.
Designation: $PA_1A_2...A_n$ .
Example: pentagonal pyramid $PA_1A_2A_3A_4A_5$ .

Triangles $PA_1A_2, \PA_2A_3$, etc. are called side faces pyramids, segments $PA_1, PA_2$, etc. – lateral ribs, polygon $A_1A_2A_3A_4A_5$ – basis, point $P$ – top.

Height pyramids are a perpendicular descended from the top of the pyramid to the plane of the base.

A pyramid with a triangle at its base is called tetrahedron.

The pyramid is called correct, if its base is a regular polygon and one of the following conditions is met:

$(a)$ the lateral edges of the pyramid are equal;

$(b)$ the height of the pyramid passes through the center of the circle circumscribed near the base;

$(c)$ the side ribs are inclined to the plane of the base at the same angle.

$(d)$ the side faces are inclined to the plane of the base at the same angle.

Regular tetrahedron- This triangular pyramid, all faces of which are equal equilateral triangles.

Theorem

Conditions $(a), (b), (c), (d)$ are equivalent.

Proof

Let's find the height of the pyramid $PH$ . Let $\alpha$ be the plane of the base of the pyramid.

1) Let us prove that $(a)$ implies $(b)$ . Let $PA_1=PA_2=PA_3=...=PA_n$ .

Because $PH\perp \alpha$, then $PH$ is perpendicular to any line lying in this plane, which means the triangles are right-angled. This means that these triangles are equal in common leg $PH$ and hypotenuse $PA_1=PA_2=PA_3=...=PA_n$ . This means $A_1H=A_2H=...=A_nH$ . This means that the points $A_1, A_2, ..., A_n$ are at the same distance from the point $H$, therefore, they lie on the same circle with the radius $A_1H$ . This circle, by definition, is circumscribed about the polygon $A_1A_2...A_n$ .

2) Let us prove that $(b)$ implies $(c)$ .

$PA_1H, PA_2H, PA_3H,..., PA_nH$ rectangular and equal on two legs. This means that their angles are also equal, therefore, $\angle PA_1H=\angle PA_2H=...=\angle PA_nH$.

3) Let us prove that $(c)$ implies $(a)$ .

Similar to the first point, triangles $PA_1H, PA_2H, PA_3H,..., PA_nH$ rectangular and along the leg and sharp corner. This means that their hypotenuses are also equal, that is, $PA_1=PA_2=PA_3=...=PA_n$ .

4) Let us prove that $(b)$ implies $(d)$ .

Because in a regular polygon the centers of the circumscribed and inscribed circles coincide (generally speaking, this point is called the center of a regular polygon), then $H$ is the center of the inscribed circle. Let's draw perpendiculars from the point $H$ to the sides of the base: $HK_1, HK_2$, etc. These are the radii of the inscribed circle (by definition). Then, according to the TTP ($PH$ is a perpendicular to the plane, $HK_1, HK_2$, etc. are projections, perpendicular to the sides) oblique $PK_1, PK_2$, etc. perpendicular to the sides $A_1A_2, A_2A_3$, etc. respectively. So, by definition $\angle PK_1H, \angle PK_2H$ equal to the angles between the side faces and the base. Because triangles $PK_1H, PK_2H, ...$ are equal (as rectangular on two sides), then the angles $\angle PK_1H, \angle PK_2H, ...$ are equal.

5) Let us prove that $(d)$ implies $(b)$ .

Similar to the fourth point, the triangles $PK_1H, PK_2H, ...$ are equal (as rectangular along the leg and acute angle), which means the segments $HK_1=HK_2=...=HK_n$ are equal. This means, by definition, $H$ is the center of a circle inscribed in the base. But because For regular polygons, the centers of the inscribed and circumscribed circles coincide, then $H$ is the center of the circumscribed circle. Chtd.

Consequence

The lateral faces of a regular pyramid are equal isosceles triangles.

Definition

The height of the lateral face of a regular pyramid drawn from its vertex is called apothem.
The apothems of all lateral faces of a regular pyramid are equal to each other and are also medians and bisectors.

Important Notes

1. The height of a regular triangular pyramid falls at the point of intersection of the heights (or bisectors, or medians) of the base (the base is a regular triangle).

2. Height is correct quadrangular pyramid falls at the point of intersection of the diagonals of the base (the base is a square).

3. The height of a regular hexagonal pyramid falls at the point of intersection of the diagonals of the base (the base is a regular hexagon).

4. The height of the pyramid is perpendicular to any straight line lying at the base.

Definition

The pyramid is called rectangular, if one of its side edges is perpendicular to the plane of the base.

Important Notes

1. In a rectangular pyramid, the edge perpendicular to the base is the height of the pyramid. That is, $SR$ is the height.

2. Because $SR$ is perpendicular to any line from the base, then $\triangle SRM, \triangle SRP$– right triangles.

3. Triangles $\triangle SRN, \triangle SRK$- also rectangular.
That is, any triangle formed by this edge and the diagonal emerging from the vertex of this edge lying at the base will be rectangular.

\[(\Large(\text(Volume and surface area of the pyramid)))\]

Theorem

The volume of the pyramid is equal to one third of the product of the area of the base and the height of the pyramid: \

Consequences

Let $a$ be the side of the base, $h$ be the height of the pyramid.

1. The volume of a regular triangular pyramid is $V_(\text(right triangle.pir.))=\dfrac(\sqrt3)(12)a^2h$,

2. The volume of a regular quadrangular pyramid is $V_(\text(right.four.pir.))=\dfrac13a^2h$.

3. The volume of a regular hexagonal pyramid is $V_(\text(right.six.pir.))=\dfrac(\sqrt3)(2)a^2h$.

4. The volume of a regular tetrahedron is $V_(\text(right tetr.))=\dfrac(\sqrt3)(12)a^3$.

Theorem

The area of the lateral surface of a regular pyramid is equal to the half product of the perimeter of the base and the apothem.

\[(\Large(\text(Frustum)))\]

Definition

Consider an arbitrary pyramid $PA_1A_2A_3...A_n$ . Let us draw a plane parallel to the base of the pyramid through a certain point lying on the side edge of the pyramid. This plane will split the pyramid into two polyhedra, one of which is a pyramid ($PB_1B_2...B_n$), and the other is called truncated pyramid($A_1A_2...A_nB_1B_2...B_n$ ).

The truncated pyramid has two bases - polygons $A_1A_2...A_n$ and $B_1B_2...B_n$ which are similar to each other.

The height of a truncated pyramid is a perpendicular drawn from some point of the upper base to the plane of the lower base.

Important Notes

1. All lateral faces of a truncated pyramid are trapezoids.

2. The segment connecting the centers of the bases of a regular truncated pyramid (that is, a pyramid obtained by cross-section of a regular pyramid) is the height.

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 in a pyramid all lateral edges have equal lengths, then the top of the pyramid is projected into the center of the 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.

Grounds 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 lateral rib to the base plane.

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

The pyramid is correct, meaning at the base equilateral triangle and all 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 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. By the theorem on the area of orthogonal projection flat figure we get:

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