Question

A cone, a hemisphere and a cylinder stand on equal bases of radius $r$ and have equal heights $h$.Prove that their volumes are in the ratio $1:2:3$.

Answer 1

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. $r=h$

$$\begin{gathered} Volumeofcone:Volumeofhemisphere:Volumeofcylinder \\ \Rightarrow \frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}:\frac{2}{3}{\mathrm{\pi r}}^3:{\mathrm{\pi r}}^2\mathrm{h} \\ \Rightarrow \frac{1}{3}:\frac{2}{3}:1 \\ \Rightarrow 1:2:3 \end{gathered}$$

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