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.

apotheme_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: \(S = \sqrt{h^2 + \left( \dfrac{c}{2\tan{(\frac{180}{n})}} \right)^2}\).