Tutorial Frustum of a Cone


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 ABCD in figure is a frustum of the cone ABE.

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.



l, being the slant height of frustum, R and r 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



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