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
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.
Height of cone
Height of the frustum
Volume of the frustum
Now the volume of the frustum cone
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.
(1) Total surface area of the frustum of a cone
(2) To find the slant height of the cone, use the Pythagorean theorem.
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.
Lateral surface area
Total surface area