Surface of a Cone (C)

A right circular cone is a cone whose base is a disk and whose vertex lies on the line perpendicular to the base and passing through the center of the base. Such a cone is characterized by the radius of the base and the altitude of the cone, that is, the distance from the vertex to the center of the base. The slant height of such a cone is the length of a straight line drawn from any point on the perimeter of the cone to the vertex. If the radius of the base is R and the altitude of the cone is H, then the slant height is

$$S \equiv {\rm Slant\;height} = \sqrt{R^2 + H^2}$$

which can be seen from the Pythagorean Theorem. The surface area of the cone (neglecting the area of the base) is given by

$$SA = {\rm Surface\;Area} = \pi R S = \pi R\sqrt{R^2 + H^2}$$

as can be seen by slicing the cone up the side and unrolling into a sector of a circle whose radius is the slant height and whose perimeter is $2\pi R$.

Exercises

1.

Find the slant height and surface area of a right circular cone whose altitude is 12 and whose base radius is 5

2.

What is the effect on the surface area of the cone of doubling the altitude and the radius of the base?

3.

What is the surface area of a cone if the altitude is the same as the radius of the base?