Slant height

The slant height of a right circular cone is the distance from any point on the circle to the apex of the cone.

Formula

The slant height of a cone is given by the formula $\sqrt{r^2+h^2}$ , where $r$ is the radius of the circle and $h$ is the height of 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 $d=\sqrt{r^2+h^2}$ 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:

The vertical height (or altitude) which is the perpendicular distance from the top down to the base.
The radius of the circular base
The slant height which is the distance from the top, down the side, to a point on the base circumference.

References

Slant height of a right cone at Math Open Reference

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