Correct 4-sided pyramid of the formula. The volume of the quadrangular pyramid. Collection and use of personal information

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Definition 1... A pyramid is called regular if its base is a regular polygon, while the top of such a pyramid is projected to 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 the side face drawn from its vertex is called apothem... In the figure it is designated as a 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)
• A 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 the pyramid is the section through the top and diagonal of the base (AOC, BOD)
• A polygon to which the top of the pyramid does not belong is called base of the pyramid(ABCD)

If at the bottom correct pyramid lies a triangle, quadrangle, etc. then it is 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 must know this initially.

• side ribs are equal between themselves
• apothems are equal
• side faces are equal each other (in this case, respectively, their areas, sides and bases are equal), that is, they are equal triangles
• all side faces are equal 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 apex of the pyramid is π, and each of them, respectively, π / n, where n is the number of sides of the base polygon
• the lateral surface area of ​​a regular pyramid is equal to half the product of the base perimeter and the apothem
• a circle can be described near the base of a regular pyramid (see also the radius of the circumscribed circle of a triangle)
• all side faces form equal angles with the base plane of the regular pyramid
• all the 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 dividing the entire volumetric figure into separate flat figures and applying their properties to find the individual elements of the pyramid, since many of the elements are common to several shapes.

It is necessary to break the whole volumetric figure into separate elements - triangles, squares, segments. Further, to apply the knowledge from the planimetry course to individual elements, which greatly simplifies the finding of the answer.

Formulas for the correct pyramid

Formulas for finding the volume and area of ​​the lateral surface:

Designations:
V - the 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 - the length of the side rib
α - flat angle at the top of the pyramid

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

, where

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

Correct 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 the truncated pyramid is one of its bases.

The height of the side face (which is an isosceles trapezoid) is called - apothem of the 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 the truncated pyramid is called the height of the truncated pyramid
• Everything faces of a regular truncated pyramid are isosceles (isosceles) trapezoids

Notes (edit)

See also: special cases (formulas) for the correct pyramid:

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

• apothem- side edge height correct pyramid, which is drawn from its vertex (in addition, the apothem is the length of the perpendicular, which is lowered from the middle of a regular polygon to 1 of its sides);
• side faces (ASB, BSC, CSD, DSA) - triangles that converge at the vertex;
• side ribs ( AS , BS , CS , DS ) - common sides of the side faces;
• top of the pyramid (t. S) - a point that connects the side edges and that does not lie in the plane of the base;
• height ( SO ) - a segment of the perpendicular, which is 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- section of the pyramid, which passes through the top and the diagonal of the base;
• base (ABCD) - a polygon that does not belong to the top of the pyramid.

Pyramid properties.

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

• near the base of the pyramid is easy to describe circle, while the top of the pyramid will be projected to the center of this circle;
• lateral ribs form the same with the base plane corners ;
• moreover, the converse is also true, i.e. when the side edges form equal angles with the base plane, or when a circle can be described near the base of the pyramid and the top of the pyramid is projected to the center of this circle, then all the side edges of the pyramid have the same size.

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

• it is easy to describe a circle near the base of the pyramid, while 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 lateral surface area is equal to ½ of the product of the base perimeter by the height of the lateral face.

3. Near the pyramid can be described scope in the event that at the base of the pyramid lies a polygon around which a circle can be described (necessary and sufficient condition). The center of the sphere will be the point of intersection of the planes that pass through the midpoints of the pyramid's edges 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 the pyramid if the bisector planes of the inner 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.

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

The pyramid will triangular, quadrangular, and so on, when the base of the pyramid is a triangle, a quadrangle, and so on. The triangular pyramid is a tetrahedron - tetrahedron... Quadrangular - pentahedron and so on.