Question

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

Answer 1

Let $r$ and $h$ be the radius of base and height of the cylinder, cone, and hemisphere.

Given,

Height of cone = height of hemisphere = height of cylinder = $h$

Radius of cone = radius of hemisphere = radius of cylinder = $r$

We know that, Height of hemisphere = Radius of the hemisphere
$\therefore$ $h=r$

So,
Volume of the cylinder $=\pi r^{2}h=\pi r^2\times r=\pi r^3$
Volume of cone $=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi r^2\times r=\frac{1}{3}\pi r^3$
Volume of hemisphere $=\frac{2}{3}\pi r^3$
Volume of cone : Volume of hemisphere : Volume of cylinder
$=\frac{1}{3}\pi r^3:\frac{2}{3}\pi r^3:\pi r^3$
$=\frac{1}{3}:\frac{2}{3}:1$ (Dividing by $\pi r^3$)
$=1:2:3$ (Multiplying by 3)
Thus, the ratio of their volumes is 1 : 2 : 3