A cylinder, a cone and a hemisphere


A cylinder, a cone and a hemisphere are of equal base and have the same height. What is the ratio of their volumes?


Let the diameter of the base for all three be x cm and height be y cm.

For hemisphere radius $=\frac{x}{2} \mathrm{~cm}$

Height $=y=\frac{x}{2} \mathrm{~cm}$

(As height of the hemisphere is equal to the radius of hemisphere)

For cone

Radius $=\frac{x}{2} \mathrm{~cm}$

Height $=\frac{x}{2} \mathrm{~cm}$

(As height is same for all)

For cylinder

Radius $=\frac{x}{2} \mathrm{~cm}$

Height $=\frac{x}{2} \mathrm{~cm}$

The ratio of their volume is

= cylinder volume : conic volume : hemispherical volume

$=\pi\left(\frac{x}{2}\right)^{2} \frac{x}{2}: \frac{1}{3} \pi\left(\frac{x}{2}\right)^{2}\left(\frac{x}{2}\right): \frac{2}{3} \pi\left(\frac{x}{3}\right)^{3}$

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


$=3: 1: 2$




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