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

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Solution

Volume of cone $= \frac{1}{3} π r^{2} h$
Volume of hemisphere $= \frac{2}{3} π r^{3}$
Volume of cylinder $= π r^{2} h$
Given that cone, hemisphere and cylinder have equal base and same height
That is $r = h$
Volume of cone $:$ Volume of hemisphere $:$ Volume of cylinder $= \frac{1}{3} π r^{2} h : \frac{2}{3} π r^{3} : π r^{2} h$
$= \frac{1}{3} π r^{3} : \frac{2}{3} π r^{3} : π r^{3}$
$= \frac{1}{3} : \frac{2}{3} : 1$
$= 1 : 2 : 3$
Hence proved.

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