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

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

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 side surface of the pyramid is:

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 main

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

Pyramid Volume

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

Proof. We will proceed from triangular prism. 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 decompose the remaining part of the prism into the core of the body by drawing a plane through the diagonals A "C" and "B" C of the side faces. The resulting two bodies are also pyramids. Considering the triangle A"B"C" as the base of one of them, and C its top, we will 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 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 BC "and B" CC "for their bases. Pyramids CA" B "C" and A "B "VS 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 to each other, the volume of each of them is equal to one third of the volume of the prism. Since the shape of the base is insignificant, 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

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Definition 1. A pyramid is called regular if its base is a regular polygon, and the top 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
The point connecting the side edges and not lying in the plane of the base is called top of the pyramid(O)
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 segment of the perpendicular 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 top and the diagonal of the base (AOC, BOD)
A polygon that does not have a pyramid vertex is called the base of the pyramid(ABCD)

If at the base correct 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 very beginning.

side ribs are equal between themselves
apothems are equal
side faces are equal among themselves (at the same time, their areas, sides and bases are equal, respectively), that is, they are equal triangles
all side faces are congruent isosceles triangles
in any regular pyramid, you can both inscribe 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 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 near the base of a regular pyramid (see also the radius of the circumscribed circle of a triangle)
all side faces form a regular pyramid with the base plane equal angles
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 is reduced to breaking the entire three-dimensional figure into separate flat figures and applying 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 separate elements - triangles, squares, segments. Further, to apply knowledge from the planimetry course to individual elements, which greatly simplifies finding the answer.

Formulas for the correct pyramid

Formulas for finding volume and lateral surface area:

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

This formula for finding volume can be used only for correct pyramid:

, where

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

Correct truncated pyramid

If you make a cut, base parallel pyramids, 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 correct if the pyramid from which it was obtained is correct.

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

Notes

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

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

Pyramid- This is a polyhedral figure, at the base of which lies a polygon, and the remaining faces are represented by triangles with a common vertex.

If the base is a square, then a pyramid is called quadrangular, if the triangle is triangular. The height of the pyramid is drawn from its top perpendicular to the base. Also used to calculate the area apothem is the height of the side face lowered from its vertex.
The formula for the area of the lateral surface of a pyramid is the sum of the areas of its lateral faces, which are equal to each other. However, this method of calculation is used very rarely. Basically, the area of \u200b\u200bthe pyramid is calculated through the perimeter of the base and the apothem:

Consider an example of calculating the area of the lateral surface of a pyramid.

Let a pyramid be given with base ABCDE and apex F. AB=BC=CD=DE=EA=3 cm. Apothem a = 5 cm. Find the area of the lateral surface of the pyramid.
Let's find the perimeter. Since all the faces of the base are equal, then the perimeter of the pentagon will be equal to:
Now you can find the side area of the pyramid:

Area of a regular triangular pyramid

A regular triangular pyramid consists of a base containing right triangle and three side faces that are equal in area.
The formula for the lateral surface area of a regular triangular pyramid can be calculated in many ways. You can apply the usual formula for calculating through the perimeter and apothem, or you can find the area of \u200b\u200bone face and multiply it by three. Since the face of the pyramid is a triangle, we apply the formula for the area of a triangle. It will require an apothem and the length of the base. Consider an example of calculating the lateral surface area of a regular triangular pyramid.

Given a pyramid with an apothem a = 4 cm and a base face b = 2 cm. Find the area of the lateral surface of the pyramid.
First, find the area of one of the side faces. In this case it will be:
Substitute the values in the formula:
Since in a regular pyramid all sides are the same, the area of the side surface of the pyramid will be equal to the sum of the areas of the three faces. Respectively:

The area of the truncated pyramid

truncated A pyramid is a polyhedron formed by a pyramid and its section parallel to the base.
The formula for the lateral surface area of a truncated pyramid is very simple. The area is equal to the product of half the sum of the perimeters of the bases and the apothem:

Consider an example of calculating the area of the lateral surface of a truncated pyramid.

Given a regular quadrangular pyramid. The lengths of the base are b = 5 cm, c = 3 cm. Apothem a = 4 cm. Find the area of the lateral surface of the figure.
First, find the perimeter of the bases. V greater ground it will be equal to:
In a smaller base:
Let's calculate the area: