Question

A cone, hemisphere and a cylinder stand on the same base and have equal height. Find the ratio of their volumes.

Answer 1

Curved surface area 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

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

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

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

$=1:2:3$