A cylinder, a hemisphere and a cone have equal base diameters and have the same height. Prove that their volumes are in the ratio 3 : 2 : 1.

Let the common radius of shapes be r and height be h.

Ratio of their volumes = Volume of cylinder : Volume of hemisphere : Volume of cone

Since, a cylinder, a hemisphere and a cone have equal base diameters and have the same height.

⇒ They share a same radius r.

For hemisphere, the height is the distance from the centre of its base to its heighest point, which is equal to the radius.

⇒ h = r

$= π\text{r}^2\text{h} : \dfrac{2}{3} π\text{r}^3 : \dfrac{1}{3} π\text{r}^2\text{h} \\[1em] = π\text{r}^2\text{h} : \dfrac{2}{3} π\text{r}^2 \text{r} : \dfrac{1}{3} π\text{r}^2\text{h} \\[1em] = π\text{r}^2\text{h} : \dfrac{2}{3} π\text{r}^2 \text{h} : \dfrac{1}{3} π\text{r}^2\text{h} \\[1em] = 1 : \dfrac{2}{3} : \dfrac{1}{3}$

On multiplying by 3, ratio = 3 : 2 : 1

Hence, proved that their volumes are in the ratio 3 : 2 : 1.