A cone, a hemisphere and a cylinder stand on equal bases of radius and have equal heights .Prove that their volumes are in the ratio .

 Solution

A cone, a hemisphere and a cylinder stand on equal bases of radius $r$ and have equal heights $h$.

We have to prove that their volumes are in the ratio $1 : 2 : 3$.

From the formula, we have learnt that

$$\begin{gathered} \mathrm{Volume} \mathrm{of} \mathrm{cylinder} = {\mathrm{\pi r}}^{2}h \\ \mathrm{Volume} \mathrm{of} \mathrm{cone} = \frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h} \\ \mathrm{Volume} \mathrm{of} \mathrm{hemisphere} = \frac{2}{3}{\mathrm{\pi r}}^3 \end{gathered}$$

Given that the cone, hemisphere and cylinder have an equal base and same height
i.e. $r=h$

$$\begin{gathered} Volume of cone : Volume of hemisphere : Volume of cylinder \\ \Rightarrow \frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h} : \frac{2}{3}{\mathrm{\pi r}}^3 : {\mathrm{\pi r}}^2\mathrm{h} \\ \Rightarrow \frac{1}{3} : \frac{2}{3} : 1 \\ \Rightarrow 1 : 2 : 3 \end{gathered}$$

Hence, the ratio of the volume of the cone, hemisphere and cylinder is $1 : 2 : 3$.

Mathematics