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Definition 1. A pyramid is called regular if its base is a regular polygon, and the vertex of such a pyramid is projected into the center of its base.

Definition 2. A pyramid is called regular if its base is a regular polygon and its height passes through the center of the base.

Elements of a regular pyramid

The height of a side face drawn from its vertex is called apothem. In the figure it is designated as segment ON
A point connecting the lateral edges and not lying in the plane of the base is called the top of the pyramid(ABOUT)
Triangles that have a common side with the base and one of the vertices coinciding with the vertex are called side faces(AOD, DOC, COB, AOB)
The perpendicular segment drawn through the top of the pyramid to the plane of its base is called pyramid height(OK)
Diagonal section of a pyramid- this is the section passing through the apex and diagonal of the base (AOC, BOD)
A polygon that does not belong to a vertex of the pyramid is called base of the pyramid(ABCD)

If at the base regular pyramid lies a triangle, quadrilateral, etc. then it's called regular triangular , quadrangular etc.

A triangular pyramid is a tetrahedron - a tetrahedron.

Properties of a regular pyramid

To solve problems, it is necessary to know the properties of individual elements, which are usually omitted in the condition, since it is believed that the student should know this from the beginning.

side ribs are equal between themselves
apothems are equal
side faces are equal among themselves (in this case, their areas, sides and bases are respectively equal), that is, they are equal triangles
all lateral faces are equal isosceles triangles
in any regular pyramid you can both fit and describe a sphere around it
if the centers of the inscribed and circumscribed spheres coincide, then the sum of the plane angles at the top of the pyramid is equal to π, and each of them is π/n, respectively, where n is the number of sides of the base polygon
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
a circle can be circumscribed around the base of a regular pyramid (see also circumscribed circle radius of a triangle)
all lateral faces form equal angles with the plane of the base of a regular pyramid
all heights of the side faces are equal to each other

Instructions for solving problems. The properties listed above should help in a practical solution. If you need to find the angles of inclination of the faces, their surface, etc., then the general technique comes down to dividing the entire volumetric figure into separate flat figures and using their properties to find individual elements of the pyramid, since many elements are common to several figures.

It is necessary to break the entire three-dimensional figure into individual elements - triangles, squares, segments. Next, apply knowledge from the planimetry course to individual elements, which greatly simplifies finding the answer.

Formulas for a regular pyramid

Formulas for finding volume and lateral surface area:

Designations:
V - volume of the pyramid
S - base area
h - height of the pyramid
Sb - lateral surface area
a - apothem (not to be confused with α)
P - base perimeter
n - number of sides of the base
b - side rib length
α - flat angle at the top of the pyramid

This formula for finding volume can be applied only For correct pyramid:

, Where

V - volume of a regular pyramid
h - height of a regular pyramid
n is the number of sides of a regular polygon, which is the base of a regular pyramid
a - side length of a regular polygon

Regular truncated pyramid

If we draw a section parallel to the base of the pyramid, then the body enclosed between these planes and the lateral surface is called truncated pyramid. This section for a truncated pyramid is one of its bases.

The height of the side face (which is an isosceles trapezoid) is called - apothem of a regular truncated pyramid.

A truncated pyramid is called regular if the pyramid from which it was derived is regular.

The distance between the bases of a truncated pyramid is called height of a truncated pyramid
All faces of a regular truncated pyramid are isosceles trapezoids

Notes

See also: special cases (formulas) for a regular pyramid:

How to use the theoretical materials provided here to solve your problem:

Surface area of the pyramid. In this article we will look at problems with regular pyramids. Let me remind you that a regular pyramid is a pyramid whose base is a regular polygon, the top of the pyramid is projected into the center of this polygon.

The side face of such a pyramid is an isosceles triangle.The altitude of this triangle drawn from the vertex of a regular pyramid is called apothem, SF - apothem:

In the type of problem presented below, you need to find the surface area of the entire pyramid or the area of its lateral surface. The blog has already discussed several problems with regular pyramids, where the question was about finding the elements (height, base edge, side edge).

Unified State Examination tasks usually examine regular triangular, quadrangular and hexagonal pyramids. I haven’t seen any problems with regular pentagonal and heptagonal pyramids.

The formula for the area of the entire surface is simple - you need to find the sum of the area of the base of the pyramid and the area of its lateral surface:

Let's consider the tasks:

The sides of the base of a regular quadrangular pyramid are 72, the side edges are 164. Find the surface area of this pyramid.

The surface area of the pyramid is equal to the sum of the areas of the lateral surface and the base:

*The lateral surface consists of four triangles of equal area. The base of the pyramid is a square.

We can calculate the area of the side of the pyramid using:

Thus, the surface area of the pyramid is:

Answer: 28224

The sides of the base of a regular hexagonal pyramid are equal to 22, the side edges are equal to 61. Find the lateral surface area of this pyramid.

The base of a regular hexagonal pyramid is a regular hexagon.

The lateral surface area of this pyramid consists of six areas of equal triangles with sides 61,61 and 22:

Let's find the area of the triangle using Heron's formula:

Thus, the lateral surface area is:

Answer: 3240

*In the problems presented above, the area of the side face could be found using another triangle formula, but for this you need to calculate the apothem.

27155. Find the surface area of a regular quadrangular pyramid whose base sides are 6 and whose height is 4.

In order to find the surface area of the pyramid, we need to know the area of the base and the area of the lateral surface:

The area of the base is 36 since it is a square with side 6.

The lateral surface consists of four faces, which are equal triangles. In order to find the area of such a triangle, you need to know its base and height (apothem):

*The area of a triangle is equal to half the product of the base and the height drawn to this base.

The base is known, it is equal to six. Let's find the height. Consider a right triangle (highlighted in yellow):

One leg is equal to 4, since this is the height of the pyramid, the other is equal to 3, since it is equal to half the edge of the base. We can find the hypotenuse using the Pythagorean theorem:

This means that the area of the lateral surface of the pyramid is:

Thus, the surface area of the entire pyramid is:

Answer: 96

27069. The sides of the base of a regular quadrangular pyramid are equal to 10, the side edges are equal to 13. Find the surface area of this pyramid.

27070. The sides of the base of a regular hexagonal pyramid are equal to 10, the side edges are equal to 13. Find the lateral surface area of this pyramid.

There are also formulas for the lateral surface area of a regular pyramid. In a regular pyramid, the base is an orthogonal projection of the lateral surface, therefore:

P- base perimeter, l- apothem of the pyramid

*This formula is based on the formula for the area of a triangle.

If you want to learn more about how these formulas are derived, don’t miss it, follow the publication of articles.That's all. Good luck to you!

Sincerely, Alexander Krutitskikh.

P.S: I would be grateful if you tell me about the site on social networks.

Before studying questions about this geometric figure and its properties, you should understand some terms. When a person hears about a pyramid, he imagines huge buildings in Egypt. This is what the simplest ones look like. But they come in different types and shapes, which means the calculation formula for geometric shapes will be different.

Types of figure

Pyramid - geometric figure, denoting and representing several faces. In essence, this is the same polyhedron, at the base of which lies a polygon, and on the sides there are triangles that connect at one point - the vertex. The figure comes in two main types:

correct;
truncated.

In the first case, the base is a regular polygon. Here all lateral surfaces are equal between themselves and the figure itself will please the eye of a perfectionist.

In the second case, there are two bases - a large one at the very bottom and a small one between the top, repeating the shape of the main one. In other words, a truncated pyramid is a polyhedron with a cross section formed parallel to the base.

Terms and symbols

Key terms:

Regular (equilateral) triangle- a figure with three equal angles and equal sides. In this case, all angles are 60 degrees. The figure is the simplest of regular polyhedra. If this figure lies at the base, then such a polyhedron will be called regular triangular. If the base is a square, the pyramid will be called a regular quadrangular pyramid.
Vertex– the highest point where the edges meet. The height of the apex is formed by a straight line extending from the apex to the base of the pyramid.
Edge– one of the planes of the polygon. It can be in the form of a triangle in the case of a triangular pyramid, or in the form of a trapezoid for a truncated pyramid.
Section- a flat figure formed as a result of dissection. It should not be confused with a section, since a section also shows what is behind the section.
Apothem- a segment drawn from the top of the pyramid to its base. It is also the height of the face where the second height point is located. This definition is valid only in relation to a regular polyhedron. For example, if this is not a truncated pyramid, then the face will be a triangle. In this case, the height of this triangle will become the apothem.

Area formulas

Find the lateral surface area of the pyramid any type can be done in several ways. If the figure is not symmetrical and is a polygon with different sides, then in this case it is easier to calculate the total surface area through the totality of all surfaces. In other words, you need to calculate the area of each face and add them together.

Depending on what parameters are known, formulas for calculating a square, trapezoid, arbitrary quadrilateral, etc. may be required. The formulas themselves in different cases will also have differences.

In the case of a regular figure, finding the area is much easier. It is enough to know just a few key parameters. In most cases, calculations are required specifically for such figures. Therefore, the corresponding formulas will be given below. Otherwise, you would have to write everything out over several pages, which would only confuse and confuse you.

Basic formula for calculation The lateral surface area of a regular pyramid will have the following form:

S=½ Pa (P is the perimeter of the base, and is the apothem)

Let's look at one example. The polyhedron has a base with segments A1, A2, A3, A4, A5, and all of them are equal to 10 cm. Let the apothem be equal to 5 cm. First you need to find the perimeter. Since all five faces of the base are the same, you can find it like this: P = 5 * 10 = 50 cm. Next, we apply the basic formula: S = ½ * 50 * 5 = 125 cm squared.

Lateral surface area of a regular triangular pyramid easiest to calculate. The formula looks like this:

S =½* ab *3, where a is the apothem, b is the face of the base. The factor of three here means the number of faces of the base, and the first part is the area of the side surface. Let's look at an example. Given a figure with an apothem of 5 cm and a base edge of 8 cm. We calculate: S = 1/2*5*8*3=60 cm squared.

Lateral surface area of a truncated pyramid It's a little more difficult to calculate. The formula looks like this: S =1/2*(p_01+ p_02)*a, where p_01 and p_02 are the perimeters of the bases, and is the apothem. Let's look at an example. Let’s say that for a quadrangular figure the dimensions of the sides of the bases are 3 and 6 cm, and the apothem is 4 cm.

Here, first you need to find the perimeters of the bases: р_01 =3*4=12 cm; р_02=6*4=24 cm. It remains to substitute the values into the main formula and we get: S =1/2*(12+24)*4=0.5*36*4=72 cm squared.

Thus, you can find the lateral surface area of a regular pyramid of any complexity. You should be careful and not confuse these calculations with the total area of the entire polyhedron. And if you still need to do this, just calculate the area of the largest base of the polyhedron and add it to the area of the lateral surface of the polyhedron.

Video

This video will help you consolidate information on how to find the lateral surface area of different pyramids.

Triangular pyramid is a polyhedron whose base is a regular triangle.

In such a pyramid, the edges of the base and the edges of the sides are equal to each other. Accordingly, the area of the side faces is found from the sum of the areas of three identical triangles. You can find the lateral surface area of a regular pyramid using the formula. And you can make the calculation several times faster. To do this, you need to apply the formula for the area of the lateral surface of a triangular pyramid:

where p is the perimeter of the base, all sides of which are equal to b, a is the apothem lowered from the top to this base. Let's consider an example of calculating the area of a triangular pyramid.

Problem: Let a regular pyramid be given. The side of the triangle at the base is b = 4 cm. The apothem of the pyramid is a = 7 cm. Find the area of the lateral surface of the pyramid.
Since, according to the conditions of the problem, we know the lengths of all the necessary elements, we will find the perimeter. We remember that in a regular triangle all sides are equal, and, therefore, the perimeter is calculated by the formula:

Let's substitute the data and find the value:

Now, knowing the perimeter, we can calculate the lateral surface area:

To apply the formula for the area of a triangular pyramid to calculate the full value, you need to find the area of the base of the polyhedron. To do this, use the formula:

The formula for the area of the base of a triangular pyramid may be different. It is possible to use any calculation of parameters for a given figure, but most often this is not required. Let's consider an example of calculating the area of the base of a triangular pyramid.

Problem: In a regular pyramid, the side of the triangle at the base is a = 6 cm. Calculate the area of the base.
To calculate, we only need the length of the side of the regular triangle located at the base of the pyramid. Let's substitute the data into the formula:

Quite often you need to find the total area of a polyhedron. To do this, you will need to add up the area of the side surface and the base.

Let's consider an example of calculating the area of a triangular pyramid.

Problem: Let a regular triangular pyramid be given. The base side is b = 4 cm, the apothem is a = 6 cm. Find the total area of the pyramid.
First, let's find the area of the lateral surface using the already known formula. Let's calculate the perimeter:

Substitute the data into the formula:
Now let's find the area of the base:
Knowing the area of the base and lateral surface, we find the total area of the pyramid:

When calculating the area of a regular pyramid, you should not forget that the base is a regular triangle and many elements of this polyhedron are equal to each other.

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

As for the total surface area, we simply add the base area to the side one.

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

Proof:

If the side of the base is a, the number of sides is n, then the lateral surface of the pyramid is equal to:

a l n/2 =a n l/2=pl/2

where l is the apothem and p is the perimeter of the base of the pyramid. The theorem has been proven.

This formula reads like this:

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.

The total surface area of the pyramid is calculated by the formula:

S full = S side +S basic

If the pyramid is irregular, then its lateral surface will be equal to the sum of the areas of its lateral faces.

Volume of the pyramid

Volume pyramid is equal to one third of the product of the area of the base and the height.

Proof. We will start from a triangular prism. Let us draw a plane through the vertex A" of the upper base of the prism and the opposite edge BC of the lower base. This plane will cut off the triangular pyramid A" ABC from the prism. We will decompose the remaining part of the prism into solid bodies, drawing a plane through the diagonals A"C and B"C of the side faces. The resulting two bodies are also pyramids. Considering triangle A"B"C" to be the base of one of them, and C to be its vertex, we see that its base and height are the same as those of the first pyramid we cut off, therefore pyramids A"ABC and CA"B"C" are equal in size. In addition, both new pyramids CA"B"C" and A"B"BC are also equal in size - this will become clear if we take the triangles BBC" and B"CC" as their bases. “The suns have a common vertex A,” and their bases are located in the same plane and are equal, therefore, the pyramids are equal in size. So, the prism is decomposed into three pyramids of equal size; the volume of each of them is equal to one third of the volume of the prism. then, in general, the volume of an n-gonal pyramid is equal to one third of the volume of a prism with the same height and the same (or equal) base. Recalling the formula expressing the volume of a prism, V=Sh, we get the final result: V=1/3Sh.