A cone, a hemisphere and a cylinder stand on equal bases and have the same height. Show that their volumes are in the ratio $1 : 2 : 3$.

 Given:

A cone, a hemisphere and a cylinder stand on equal bases and have the same height. 

To do:

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

Solution:

Bases and heights of the cone, hemisphere and the cylinder are equal.

Let $r$ be the radius and $h$ be their heights.

This implies,

Volume of the cone $=\frac{1}{3} \pi r^{2} h$

Volume of the hemisphere $=\frac{2}{3} \pi r^{3}$

Volume of the cylinder $=\pi r^{2} h$

Therefore,

Ratio in their volumes $=\frac{1}{3} \pi r^{2} h: \frac{2}{3} \pi r^{3}: \pi r^{2} h$

$=\frac{1}{3} h: \frac{2}{3} r: h$

$=\frac{1}{3} h: \frac{2}{3} h: h$

$=\frac{1}{3}: \frac{2}{3}: 1$

$=1: 2: 3$

Hence proved.

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Updated on: 10-Oct-2022

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