# Frustum of a Cone

If a cone is cut by a plane parallel to its base, the portion of a solid between this plane and the base is known as the frustum of a cone.

The volume denoted by in the figure is the frustum of the cone .

__Volume of the Frustum of a Cone__

Since we know that cone is a limit of a pyramid, therefore the frustum of a cone will be the limit of the frustum of a pyramid. But the volume of a pyramid is

Where

__Example:__A cone **12cm** high is cut **8cm** from the vertex to form a frustum with a volume of **190cu.cm**. Find the radius of the cone.

__Solution__**:**

Given that

Height of cone

Height of the frustum

Volume of the frustum

Now the volume of the frustum cone

or

or

Hence the required radius of the cone

__Curved Surface Area of the Frustum of a Cone__

Since a cone is the limiting case of a pyramid, therefore the lateral surface of the frustum of a cone can be deduced from the slant surface of the frustum of a pyramid, i.e. the curved (lateral) surface of the frustum of the cone.

, being the slant height of the frustum, and being the two radii of bases.

Note:

(1) Total surface area of the frustum of a cone

(2) To find the slant height of the cone, use the Pythagorean theorem.

__Example__**:**

A bucket is in the shape of the frustum of a right circular cone, as shown in the figure below. Find the volume and the total surface area of the bucket.

__Solution__**:**

Slant height

Lateral surface area

Base areas

Total surface area

Volume