Regular triangular pyramid and its main elements. Pyramid. Visual Guide (2019)

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 side 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 regular polygon, then the bases of all lateral faces are equal to each other. Consequently, all lateral faces are equal according to the III criterion of equality of triangles.

The theorem is 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, all sides are equal, then

The theorem is 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 is 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.

Using the midline theorem, 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

  • 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 the side edges have 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 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.

This video tutorial will help users get an idea of ​​the Pyramid theme. Correct pyramid. In this lesson we will get acquainted with the concept of a pyramid and give it a definition. Let's consider what a regular pyramid is and what properties it has. Then we prove the theorem about the lateral surface of a regular pyramid.

In this lesson we will get acquainted with the concept of a pyramid and give it a definition.

Consider a polygon A 1 A 2...A n, which lies in the α plane, and the point P, which does not lie in the α plane (Fig. 1). Let's connect the dots P with vertices A 1, A 2, A 3, … A n. We get n triangles: A 1 A 2 R, A 2 A 3 R and so on.

Definition. Polyhedron RA 1 A 2 ...A n, made up of n-square A 1 A 2...A n And n triangles RA 1 A 2, RA 2 A 3 …RA n A n-1 is called n-coal pyramid. Rice. 1.

Rice. 1

Consider a quadrangular pyramid PABCD(Fig. 2).

R- the top of the pyramid.

ABCD- the base of the pyramid.

RA- side rib.

AB- base rib.

From the point R let's drop the perpendicular RN to the base plane ABCD. The perpendicular drawn is the height of the pyramid.

Rice. 2

Full surface The pyramid consists of a lateral surface, that is, the area of ​​all lateral faces, and the area of ​​the base:

S full = S side + S main

A pyramid is called correct if:

  • its base is a regular polygon;
  • the segment connecting the top of the pyramid to the center of the base is its height.

Explanation using the example of a regular quadrangular pyramid

Consider a regular quadrangular pyramid PABCD(Fig. 3).

R- the top of the pyramid. Base of the pyramid ABCD- a regular quadrilateral, that is, a square. Dot ABOUT, the point of intersection of the diagonals, is the center of the square. Means, RO is the height of the pyramid.

Rice. 3

Explanation: in the correct n In a triangle, the center of the inscribed circle and the center of the circumcircle coincide. This center is called the center of the polygon. Sometimes they say that the vertex is projected into the center.

The height of the lateral face of a regular pyramid drawn from its vertex is called apothem and is designated h a.

1. all lateral edges of a regular pyramid are equal;

2. The side faces are equal isosceles triangles.

We will give a proof of these properties using the example of a regular quadrangular pyramid.

Given: PABCD- regular quadrangular pyramid,

ABCD- square,

RO- height of the pyramid.

Prove:

1. RA = PB = RS = PD

2.∆ABP = ∆BCP =∆CDP =∆DAP See Fig. 4.

Rice. 4

Proof.

RO- height of the pyramid. That is, straight RO perpendicular to the plane ABC, and therefore direct JSC, VO, SO And DO lying in it. So triangles ROA, ROV, ROS, ROD- rectangular.

Consider a square ABCD. From the properties of a square it follows that AO = VO = CO = DO.

Then the right triangles ROA, ROV, ROS, ROD leg RO- general and legs JSC, VO, SO And DO are equal, which means that these triangles are equal on two sides. From the equality of triangles follows the equality of segments, RA = PB = RS = PD. Point 1 has been proven.

Segments AB And Sun are equal because they are sides of the same square, RA = PB = RS. So triangles AVR And VSR - isosceles and equal on three sides.

In a similar way we find that triangles ABP, VCP, CDP, DAP are isosceles and equal, as required to be proved in paragraph 2.

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

To prove this, let’s choose a regular triangular pyramid.

Given: RAVS- correct triangular pyramid.

AB = BC = AC.

RO- height.

Prove: . See Fig. 5.

Rice. 5

Proof.

RAVS- regular triangular pyramid. That is AB= AC = BC. Let ABOUT- center of the triangle ABC, Then RO is the height of the pyramid. At the base of the pyramid lies an equilateral triangle ABC. notice, that .

Triangles RAV, RVS, RSA- equal isosceles triangles (by property). A triangular pyramid has three side faces: RAV, RVS, RSA. This means that the area of ​​the lateral surface of the pyramid is:

S side = 3S RAW

The theorem is proven.

The radius of a circle inscribed at the base of a regular quadrangular pyramid is 3 m, the height of the pyramid is 4 m. Find the area of ​​the lateral surface of the pyramid.

Given: regular quadrangular pyramid ABCD,

ABCD- square,

r= 3 m,

RO- height of the pyramid,

RO= 4 m.

Find: S side. See Fig. 6.

Rice. 6

Solution.

According to the proven theorem, .

Let's first find the side of the base AB. We know that the radius of a circle inscribed at the base of a regular quadrangular pyramid is 3 m.

Then, m.

Find the perimeter of the square ABCD with a side of 6 m:

Consider a triangle BCD. Let M- middle of the side DC. Because ABOUT- middle BD, That (m).

Triangle DPC- isosceles. M- middle DC. That is, RM- median, and therefore the height in the triangle DPC. Then RM- apothem of the pyramid.

RO- height of the pyramid. Then, straight RO perpendicular to the plane ABC, and therefore direct OM, lying in it. Let's find the apothem RM from right triangle ROM.

Now we can find lateral surface pyramids:

Answer: 60 m2.

The radius of the circle circumscribed around the base of a regular triangular pyramid is equal to m. The lateral surface area is 18 m 2. Find the length of the apothem.

Given: ABCP- regular triangular pyramid,

AB = BC = SA,

R= m,

S side = 18 m2.

Find: . See Fig. 7.

Rice. 7

Solution.

In a right triangle ABC The radius of the circumscribed circle is given. Let's find a side AB this triangle using the law of sines.

Knowing the side of a regular triangle (m), we find its perimeter.

By the theorem on the lateral surface area of ​​a regular pyramid, where h a- apothem of the pyramid. Then:

Answer: 4 m.

So, we looked at what a pyramid is, what a regular pyramid is, and we proved the theorem about the lateral surface of a regular pyramid. In the next lesson we will get acquainted with the truncated pyramid.

Bibliography

  1. Geometry. Grades 10-11: textbook for students of general education institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th ed., rev. and additional - M.: Mnemosyne, 2008. - 288 p.: ill.
  2. Geometry. Grades 10-11: Textbook for general education educational institutions/ Sharygin I.F. - M.: Bustard, 1999. - 208 p.: ill.
  3. Geometry. Grade 10: Textbook for general education institutions with in-depth and specialized study of mathematics /E. V. Potoskuev, L. I. Zvalich. - 6th ed., stereotype. - M.: Bustard, 008. - 233 p.: ill.
  1. Internet portal "Yaklass" ()
  2. Internet portal “Festival of pedagogical ideas “First of September” ()
  3. Internet portal “Slideshare.net” ()

Homework

  1. Can a regular polygon be the base of an irregular pyramid?
  2. Prove that disjoint edges of a regular pyramid are perpendicular.
  3. Find the value of the dihedral angle at the side of the base of a regular quadrangular pyramid if the apothem of the pyramid is equal to the side of its base.
  4. RAVS- regular triangular pyramid. Construct the linear angle of the dihedral angle at the base of the pyramid.

In which one of the side ribs is perpendicular to the base.

In this case, this edge will be the height of the pyramid.

Properties of the pyramid.

1. When all the side edges have 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 side ribs form equal angles with the plane of the base, or when a circle can be described around the base of the pyramid and the top of the pyramid will be projected into the center of this circle, it 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.

5. The cone will be inscribed in the pyramid when their vertices coincide, and the base of the cone will be inscribed in the base of the pyramid. In this case, it is possible to fit a cone into a pyramid only if the apothems of the pyramid have equal sizes (a necessary and sufficient condition);

6. The cone will be described near the pyramid if their vertices coincide, and the base of the cone will be described near the base of the pyramid. In this case, it is possible to describe a cone near a pyramid only if all the lateral edges of the pyramid have the same values ​​(a necessary and sufficient condition). The heights of these cones and pyramids are the same.

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

8. The cylinder will be described near the pyramid when the top of the pyramid belongs to one of its bases, and the second base of the cylinder will be described near the base of the pyramid. In this case, it is possible to describe a cylinder near a pyramid only if the base of the pyramid is an inscribed polygon (a necessary and sufficient condition).

Formulas for determining the volume and area of ​​a rectangular pyramid.

V- volume of the pyramid,

S- area of ​​the base of the pyramid,

h- height of the pyramid,

Sb- area of ​​the lateral surface of the pyramid,

a- apothem (not to be confused with α ) pyramids,

P- perimeter of the base of the pyramid,

n- the number of sides of the base of the pyramid,

b- length lateral rib pyramids,

α - flat angle at the top of the pyramid.