Question

The ratio of the volume of a cylinder: volume of cone: volume of the hemisphere of same radius and same height is

• A. $1:2:3$
• B. $3:1:2$
• C. $1:1:1$
• D. $2:3:1$

Answer 1

The correct option is B

$3:1:2$

Step 1: Cylinder, Cone and Hemisphere :

Cylinder

1. The cylinder is a three-dimensional shape having a circular base.
2. The volume of the cylinder can be given by the product of the area of base and height.
3. The volume of the cylinder $={\mathrm{\pi r}}^2\mathrm{h}$

Cone

1. A cone is a pyramid with a circular cross-section.
2. The volume of the cone $=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$

Hemisphere

1. A sphere is defined as a set of points in three-dimension, and all the points lying on the surface is equidistant from the centre.
2. The volume of the hemisphere $=\frac{2}{3}{\mathrm{\pi r}}^3$

Step 2: Explanation for correct option :

Option (b): $3:1:2$

Let $r$ be radius and $h$ be the height of the cylinder, cone, and hemisphere.

Given radius and height is the same for cylinder, cone, and hemisphere.

Height of the hemisphere $=$ radius of the hemisphere

So, $r=h$

The volume of the cylinder $={\mathrm{\pi r}}^2\mathrm{h}$

$={\mathrm{\pi r}}^3$

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

$=\frac{1}{3}{\mathrm{\pi r}}^3$

The volume of the hemisphere $=\frac{2}{3}{\mathrm{\pi r}}^3$

Volume of a cylinder: Volume of cone: Volume of the hemisphere $={\mathrm{\pi r}}^3:\frac{1}{3}{\mathrm{\pi r}}^3:\frac{2}{3}{\mathrm{\pi r}}^3$

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

$=3:1:2$

Therefore, option (b) is the correct answer.

Step 3: Explanation for incorrect option :

By calculating the ratio of the volume of a cylinder: volume of cone: volume of the hemisphere of same radius and same height, we get $3:1:2$.

Therefore, option (a), (c) and (d) are incorrect answers.

Hence, option (b) is the correct answer.