In what ratio are the volumes of a cylinder, a cone and a sphere, if…

The correct answer is Option (3) → 3 : 1 : 2

A sphere’s height is its diameter, so a common diameter \(2r\) and a common height mean \(h=2r\) for all three solids.

The three volumes (with \(h=2r\)):

• Cylinder: \(\pi r^2h=\pi r^2\times2r=2\pi r^3\)

• Cone: \(\displaystyle\frac{1}{3}\pi r^2h=\frac{1}{3}\pi r^2\times2r=\frac{2}{3}\pi r^3\)

• Sphere: \(\displaystyle\frac{4}{3}\pi r^3\)

The ratio (all three volumes carry the factor \(\pi r^3\), so it drops out):

• Ratio \(\displaystyle=2:\frac{2}{3}:\frac{4}{3}\), that is 3 : 1 : 2