# 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 frustum of a cone.

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

__Volume of Frustum of a Cone__**:**

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

Volume of frustum

Now volume of frustum cone

or

or

Hence required radius of cone

__Curved Surface Area of a Frustum of a Cone__**:**

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

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

Note:

(1) Total surface area of frustum of a cone

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

__Example__**:**

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

__Solution__**:**

Slant height

Lateral surface area

Base areas

Total surface area

Volume