Mathematics

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
2 : 1 : 3
2 : 3 : 1
3 : 2 : 1

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

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

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

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

Hence, option 1 is the correct option.

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