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 the 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.
- lateral ribs equal among themselves
- apothems are equal
- side faces 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 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:
Quadrangular pyramid is a polyhedron whose base is a square, and all its side faces are identical isosceles triangles.
This polyhedron has many different properties:
- Its lateral edges and adjacent dihedral angles are equal to each other;
- The areas of the side faces are the same;
- At the base of a regular quadrangular pyramid lies a square;
- The height dropped from the top of the pyramid intersects the point where the diagonals of the base intersect.
All these properties make it easy to find. However, quite often, in addition to this, it is necessary to calculate the volume of the polyhedron. To do this, use the formula for the volume of a quadrangular pyramid:
That is, the volume of the pyramid is equal to one third of the product of the height of the pyramid and the area of the base. Since it is equal to the product of its equal sides, we immediately enter the formula for the area of a square into the expression for volume.
Let's consider an example of calculating the volume of a quadrangular pyramid.
Let a quadrangular pyramid be given, the base of which is a square with side a = 6 cm. The side face of the pyramid is b = 8 cm. Find the volume of the pyramid. 
To find the volume of a given polyhedron, we need the length of its height. Therefore, we will find it by applying the Pythagorean theorem. First, let's calculate the length of the diagonal. In the blue triangle it will be the hypotenuse. It is also worth remembering that the diagonals of a square are equal to each other and are divided in half at the point of intersection: 
Now from the red triangle we find the height h we need. It will be equal to: 
Let's substitute the necessary values and find the height of the pyramid:
Now, knowing the height, we can substitute all the values into the formula for the volume of the pyramid and calculate the required value:
In this way, knowing a few simple formulas, we were able to calculate the volume of a regular quadrangular pyramid. Remember that this value is measured in cubic units.
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- 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 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.
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.