Question

A sphere, a cylinder and a cone have the same diameter. The height of the cylinder and also the cone are equal to the diameter of the sphere. Find the ratio of their volumes.

Answer 1

Diameter of a sphere, cylinder and a cone are equal
Let each as diameter$=2r$
Then radius of each $=r$
Height of cylinder = diameter $=2r....\left(i\right)$
And height of cone $=2r$
Now volume of sphere $=\frac{4}{3}\pi r^3$
Volume of cylinder $=\pi r^{2}h$
$=\pi r^2\times 2r=2\pi r^3$ [From (i)]
and volume of cone $=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi r^2\times 2r$
$=\frac{2}{3}\pi r^3$
Now ratio of them
$\frac{4}{3}\pi r^3:2\pi r^3:\frac{2}{3}\pi r^3$
$\Rightarrow \frac{4}{3}:2:\frac{2}{3}$
$\Rightarrow 4:6:2\Rightarrow 2:3:1$
Hence, ratio of their volume are $2:3:1$