**Question:**

Find the ratio of the volumes of a cylinder, a cone and a sphere, if each has the same diameter and same height?

**Solution:**

Let the radius of the sphere be $r$.

We have,

The radius of the cone $=$ The radius of the cylinder $=$ The radius of the sphere $=r$ and

The height of the cylinder = The height of the cone = The height of the sphere $=2 r$

Now,

Volume of the cylinder $=\pi r^{2}(2 r)=2 \pi r^{3}$,

Volume of the cone $=\frac{1}{3} \pi r^{2}(2 r)=\frac{2}{3} \pi r^{3}$ and

Volume of the sphere $=\frac{4}{3} \pi r^{3}$

So.

The ratio of the volumes of the cylinder, the cone and the sphere $=2 \pi r^{3}: \frac{2}{3} \pi r^{3}: \frac{4}{3} \pi r^{3}$

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

$=3: 1: 2$

So, the ratio of the volumes of the cylinder, the cone and the sphere is 3 : 1 : 2.