Apothem in a pyramid call the piece which is carried out from its top to the basis of one of side sides if the piece is perpendicular to this basis. The side side of such volume figure always has triangular shape. Therefore in need of calculation of length of an apothem use of properties both a polyhedron (pyramid), and polygon (triangle) is admissible.

## It is required to you

- - geometrical parameters of a pyramid.

## Instruction

1. In a triangle of a side side the apothem (f) is height therefore with the known length of a side edge (b) and coal (γ) between it and an edge on which the apothem is lowered it is possible to use the known formula of calculation of height of a triangle. Increase the set edge length by a sine of the known corner: f = b*sin(γ). This formula is applicable to pyramids any (correct or wrong) forms.

2. For calculation of each of three apothems (f) of the regular triangular pyramid the nobility only one parameter - length of an edge (a) is enough. This results from the fact that sides of such pyramid have the form of equilateral triangles of the identical sizes. For finding of heights of each of them calculate a half of the work of length of an edge on a square root from three: f = a * √ 3/2.

3. If the area (s) of a side side of a pyramid is known, in addition to it it is enough to know length (a) of the general edge of this side with the basis of a volume figure. In this case you find length of an apothem (f) doubling of a ratio between the area and length of an edge: f = 2*s/a.

4. Knowing the total area of a surface of a pyramid (S) and perimeter of its basis (p) too it is possible to calculate an apothem (f), but only for a polyhedron of the correct form. Double surface area and divide result into perimeter: f = 2*S/p. The basis form in this case does not matter.

5. The quantity of tops or the parties of the basis (n) needs to be known in case in conditions length of an edge (b) of a side side and size of a corner are given (α) which is formed by two adjacent side edges of a regular pyramid. Under such initial conditions calculate an apothem (f) multiplication of number of the parties of the basis by a sine of the known corner and the squared length of a side edge with the subsequent division of the received size in half: f = n*sin (α)*b²/2.

6. In a regular pyramid with the quadrangular basis for finding of length of an apothem (f) it is possible to use height of a polyhedron (H) and length of an edge of the basis (a). Take a square root from the sum of the squared height and a quarter of the squared edge length: f = √ (H²+a²/4).