In what ratio are the volumes of a cylinder, a cone and a sphere, if each has the same diameter and the same height?

3 : 1 : 2

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

Let the diameter = d and the height = h for all three solids. Then the radius $r = \frac{d}{2}$​.

Volume of a Cylinder: $V_1 = \pi r^2 h$

Volume of a Cone: $V_2 = \frac{1}{3} \pi r^2 h$

Volume of a Sphere: 

For a sphere, the diameter = height ⇒ $h=2r$.

So,

$V_3 = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi r^2 \left(\frac{h}{2}\right)$

$V_3 = \frac{2}{3} \pi r^2 h$

Now, the ratio:

$V_{\text{cylinder}} : V_{\text{cone}} : V_{\text{sphere}} = \pi r^2 h : \frac{1}{3} \pi r^2 h : \frac{2}{3} \pi r^2 h$

Cancel out $\pi r^2 h$:

$1 : \frac{1}{3} : \frac{2}{3}$​

Multiply through by 3 to remove fractions: 3 : 1 : 2

Ratio of volumes (Cylinder : Cone : Sphere) = 3 : 1 : 2