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Pyramid
This section contains formulas for finding the height, area, volume of a pyramid (including a truncated one).
The names of its elements are described (vertex, apothem, edge, face, height, diagonal section).
In the subsections you can see examples of solving problems about pyramids.
Pyramid Volume
The volume of any pyramid (including triangular) is equal to one third of the product of the area of its base by the height
Volume of a truncated pyramid
H – truncated pyramid height; S1 – area of the base of the pyramid; S2 – area of the top base of the pyramid.
Properties of a regular pyramid
A pyramid is called regular if its base is a regular polygon, and the vertex is projected into the center of the base.
A regular pyramid has the following properties:
- side edges of a regular pyramid are equal;
- in a regular pyramid, all side faces are equal isosceles triangles;
- a sphere can be inscribed in any regular pyramid
- a sphere can be described near any regular pyramid;
- 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, 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.
How to find the area of a pyramid
The area of the pyramid (S) is equal to the sum of the area of its base (Sbase) and the side surface SSideSurface (Formula 1)
Accordingly, since the area of the lateral surface of the pyramid is equal to the sum of the surfaces of all its faces (S1,S2...Sn), we obtain formula 2.
If the pyramid is regular, then the areas of all its side faces are equal to each other.
Then it is enough to find the area of the base of the pyramid and add to it the area of the side face (SFace)
multiplied by their number (n) (see Formula 3).
Since we know that in a regular pyramid all faces are an isosceles triangle, then using the formula
for the area of an isosceles triangle, we get Formula 4 - where the area of the side surface of the pyramid
will be equal to the product of half the perimeter of the base (P) by the apothem (a)..
To find the area of a regular triangular pyramid, we use the formula for the area of an equilateral triangle with side b,
to which we add the area of three faces, which are an isosceles triangle with base b and height a
(it is also the apothem of a regular triangular pyramid). As a result, we get Formula 5.
If the pyramid is a tetrahedron, then all its faces are equal to each other and the surface area
of such a pyramid is equal to the square of the side (b) multiplied by the root of three (Formula 6).
The surface area of a regular quadrangular pyramid is found according to the general rule -
since there is a square at the base, its area is equal to the square of the base side (b),
to which the area of the four faces of the sides is added (Formula 7).
Stereometry |
Описание курса
| Regular pyramid
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