Question

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

Answer 1

$\because$ Bases and heights of a cones hemisphere and a cylinder are equal
Let r be the radius and h be their heights
now volume of cone $=\frac{1}{3}\pi r^{2}h$
Volume of Hemisphere $=\frac{2}{3}\pi r^3$
and volume of cylinder $=\pi r^{2}h$
$\therefore$ Ratio in thir volumes
$=\frac{1}{3}\pi r^{2}h:\frac{2}{3}\pi r^3:\pi r^{2}h=\frac{1}{3}h:\frac{2}{3}r:h=\frac{1}{3}h:\frac{2}{3}h:h=\frac{1}{3}:\frac{2}{3}:1=1:2:3$