Pyramid. Properties and patterns

A pyramid is a polyhedron with one base, and all side walls connect at one common point - the vertex. How do we divide the pyramids? What figure can be its basis? What is the formula for the volume and surface area of ​​a pyramid?

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1. Definition and properties of a pyramid

The pyramid is a geometric solid, a three-dimensional figure, a polyhedron. All its vertices, except one, lie in one plane, on a polygon called the base. The sides of the pyramid are the edges of the base, and the plane is the plane of the base.

The point lying outside the plane of the pyramid is called the vertex, and the segments connecting it with the vertices of the base are the side edges. Each edge of the base, together with the vertex, forms an isosceles triangle we call the side wall.

The height of the pyramid is the distance from the vertex to the plane of the base. The point which is the perpendicular projection of the pyramid's apex onto the ground plane is defined as the bottom of the height.

The base of the pyramid can be any polygon. The pyramid with a triangle at its base is a tetrahedron. The total area of ​​this solid is the sum of the areas of the pyramid's walls and the area of ​​its base.

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2. How do we divide the pyramids?

The pyramids can be divided into:

  • correct pyramid - its base is a regular polygon, hence all side edges of this solid are of equal length, all the angles of inclination of the side edges to the plane have equal measures, and the side walls are congruent isosceles triangles;
  • simple pyramid - each side edge of the pyramid is of equal length;
  • inclined pyramid - its side edges are not perpendicular to the base.

We also divide them due to the number of angles contained in the base and thus distinguish the pyramids:

  • triangular;
  • quadrangular;
  • pentagonal;
  • hexagonal.


Table of Contents...

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3. How to calculate the surface area of ​​a pyramid?

The formula for the surface area of ​​the pyramid looks like this:

Pc = Pp + Pb


  • Pp is the area of ​​the base of the pyramid;
  • Pb is the sum of the areas of the side walls of the pyramid.

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4. How to calculate the volume of the pyramid?

The formula for the volume of the pyramid is:

V = ⅓ x Pp x H.


  • Pp is the base area of ​​the pyramid;
  • H is the height of the pyramid.
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