Slant height

Formula

The slant height of a cone is given by the formula r 2 + h 2 , where r is the radius of the cirf the cone (the distance from its apex to the center of the circular base).

Derivation

If the line segment from the center of the circle to its radius is taken as one leg of a right triangle inscribed within the cone, and the second leg of the triangle runs from the apex of the cone to the center of the circle, then one leg will have length r , another leg will have length h , and by the Pythagorean theorem, r 2 + h 2 = d 2 , and d = r 2 + h 2 gives the length of the circle to the apex of the cone. This application is primarily useful in determining the slant height of a cone when given other information regarding the radius or height.

Use

The variety of geometric implications of the slant height has made it a commonly seen factor in the mathematical community for 3-d geometric study.

Relationship with height and radius

A cone is defined primarily by three central aspects, with which one can determine any one factor given the other two. They are as follows: