Question

A cone, a hemisphere and a cylinder stand on a equal bases and have the same height. Show that their volume are in the ratio $1:2:3$.

Answer 1

Volume of cone $=\frac{1}{3}\pi r^{2}h$

Volume of hemisphere $=\frac{2}{3}\pi r^3$

Volume of cylinder $=\pi r^{2}h$

Given that cone, hemisphere and cylinder have equal base and same height

That is $r=h$

Volume of cone $:$ Volume of hemisphere $:$ Volume of cylinder $=\frac{1}{3}\pi r^{2}h:\frac{2}{3}\pi r^3:\pi r^{2}h$

$=\frac{1}{3}\pi r^3:\frac{2}{3}\pi r^3:\pi r^3$

$=\frac{1}{3}:\frac{2}{3}:1$

$=1:2:3$

Hence proved.