# Slant Height of a Regular Pyramid

## Slant Height of a Regular Pyramid

Distance a from the vertex O of the pyramid to any of the edges of its base.

### Example

This distance is the length of the perpendicular segment drawn from the vertex of the regular pyramid to any of the sides of its base. ### Formula

The formula for the measure of the apothem of a regular polygon with n sides can be used to establish the relationship that provides the measure of the slant height of a right regular pyramid. In the figure above, S represents the slant height of the pyramid, h the height of the pyramid and a the apothem of the polygonal base or the radius of the circle inscribed in the base. It may be noted that since these three elements form a right triangle with a hypotenuse S, then $$S = \sqrt{h^2 + a^2}$$. By using the formula for the measure a of the apothem of the polygonal base with n sides that have a length of c, the following relationship is obtained: $$S = \sqrt{h^2 + \left( \dfrac{c}{2\tan{(\frac{180}{n})}} \right)^2}$$.