A plane parallel to a cone’s base removes a smaller similar cone. The remaining part is a frustum. Corresponding radii, perpendicular heights and slant heights have the same ratio; volumes have the cube of that ratio. Example A cone of radius 6 cm and height 12 cm is cut parallel to its base, leaving a top radius of 3 cm. The removed cone has half the original lengths, so its height is 6 cm and the frustum height is 12-6=6 cm. The frustum volume is 13pi(62)(12)- 13pi(32)(6)=126pi cm3. The original and removed slant heights are 6 5 and 3 5 cm. Frustum curved area =pi(6)(6 5)-pi(3)(3 5)=27pi 5 cm2. If both flat faces are exposed, add 62+ 32=45pi cm2. This is an application of similarity and composite-solid subtraction. Do not use the frustum’s height as the height of either complete cone; locate the common apex first.

A plane parallel to a cone’s base removes a smaller similar cone. The remaining part is a frustum. Corresponding radii, perpendicular heights and slant heights have the same ratio; volumes have the cube of that ratio.

A cone of radius 6 cm and height 12 cm is cut parallel to its base, leaving a top radius of 3 cm. The removed cone has half the original lengths, so its height is 6 cm and the frustum height is \(12-6=6\text{ cm}\). The frustum volume is \(\dfrac13\pi(6^2)(12)-\dfrac13\pi(3^2)(6)=126\pi\text{ cm}^3\).

The original and removed slant heights are \(6\sqrt5\) and \(3\sqrt5\) cm. Frustum curved area \(=\pi(6)(6\sqrt5)-\pi(3)(3\sqrt5)=27\pi\sqrt5\text{ cm}^2\). If both flat faces are exposed, add \(\pi6^2+\pi3^2=45\pi\text{ cm}^2\).

This is an application of similarity and composite-solid subtraction. Do not use the frustum’s height as the height of either complete cone; locate the common apex first.

Parallel cuts and frustums

Parallel cuts and frustums