Question

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

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{sphere}\mathrm{be}r. \\ \mathrm{We}\mathrm{have}, \\ \mathrm{The}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{cone}=\mathrm{The}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{cylinder}=\mathrm{The}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{sphere}=r\mathit{}\mathrm{and} \\ \mathrm{The}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{cylinder}=\mathrm{The}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{cone}=\mathrm{The}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{sphere}=2r \\ \mathrm{Now}, \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cylinder}=\mathrm{\pi }{r}^2\left(2r\right)=2\mathrm{\pi }{r}^3, \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cone}=\frac{1}{3}\mathrm{\pi }{r}^2\left(2r\right)=\frac{2}{3}\mathrm{\pi }{r}^3\mathrm{and} \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{sphere}=\frac{4}{3}\mathrm{\pi }{r}^3 \\ \mathrm{So}, \\ \mathrm{The}\mathrm{ratio}\mathrm{of}\mathrm{the}\mathrm{volumes}\mathrm{of}\mathrm{the}\mathrm{cylinder},\mathrm{the}\mathrm{cone}\mathrm{and}\mathrm{the}\mathrm{sphere}=2\mathrm{\pi }{r}^3:\frac{2}{3}\mathrm{\pi }{r}^3:\frac{4}{3}\mathrm{\pi }{r}^3 \\ =1:\frac{1}{3}:\frac{2}{3} \\ =3:1:2 \end{gathered}$$

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