Question

A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is __________.

Answer 1

It is given that a cone, a hemisphere and a cylinder stand on equal bases and have the same height.

Let the radius of cone, hemisphere and cylinder be r units.

Radius of the cone = Radius of hemisphere = Radius of cylinder = r

Also,

Height of the cone = Height of the cylinder = Height of the hemisphere

We know that, the height of a hemisphere is same as its radius.

∴ Height of the hemisphere = r

⇒ Height of the cone = Height of the cylinder = Height of the hemisphere = r

Now,

Volume of the cone = $\frac{1}{3}\mathrm{\pi }$ × (Radius)² × Height = $\frac{1}{3}\mathrm{\pi }$ × r² × r = $\frac{1}{3}\mathrm{\pi }$r³

Volume of the hemisphere = $\frac{2}{3}\mathrm{\pi }$ × (Radius)³ = $\frac{2}{3}\mathrm{\pi }$r³

Volume of the cylinder = $\mathrm{\pi }$ × (Radius)² × Height = $\mathrm{\pi }$ × r² × r = $\mathrm{\pi }$r³

∴ Volume of the cone : Volume of the hemisphere : Volume of the cylinder

= $\frac{1}{3}\mathrm{\pi }$r³ : $\frac{2}{3}\mathrm{\pi }$r³ : $\mathrm{\pi }$r³

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

= 1 : 2 : 3

Thus, the ratio of their volumes is 1 : 2 : 3.

A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is ____1 : 2 : 3____.