Geometry

Solids

Medium

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

1 : 3 : 2

2 : 3 : 1

3 : 1 : 2

3 : 2 : 1

✅ Correct Option: 3

All three shapes have the same diameter $d$ and the same height $h$ .

This means radius $r = \frac{d}{2}$ for all three shapes.

For a sphere, the height equals its diameter (the distance from top to bottom).

Therefore: $h = d = 2r$

The volume formulas are:

Cylinder: $V = \pi r^2 h$

Cone: $V = \frac{1}{3}\pi r^2 h$

Sphere: $V = \frac{4}{3}\pi r^3$

Substituting $h = 2r$ into each formula:

Cylinder:

$V_1 = \pi r^2 h$

$V_1 = \pi r^2(2r)$

$V_1 = 2\pi r^3$

Cone:

$V_2 = \frac{1}{3}\pi r^2 h$

$V_2 = \frac{1}{3}\pi r^2(2r)$

$V_2 = \frac{2\pi r^3}{3}$

Sphere:

$V_3 = \frac{4}{3}\pi r^3$

The ratio is:

Cylinder : Cone : Sphere $= 2\pi r^3 : \frac{2\pi r^3}{3} : \frac{4\pi r^3}{3}$

Removing the common factor $\pi r^3$ :

$= 2 : \frac{2}{3} : \frac{4}{3}$

Multiplying by 3:

$= 6 : 2 : 4$

Dividing by 2:

$= 3 : 1 : 2$

Therefore, the volume ratio is $3 : 1 : 2$ .

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2025: 15 May Shift 1

$V_1, V_2, V_3$ and $V_4$ are the volumes of four cubes of side lengths x cm, 2x cm, 3x cm and 4x cm respectively. The following statements regarding these volumes are given below.

(A) $V_1 + V_2 + 2V_3 < V_4$

(B) $V_1 + 3V_2 > V_3 + V_4$

(D) $V_1 + 4V_2 + V_3 < V_4$

Which of these statements is/are correct?

Choose the correct answer from the options given below:

2022: 26 Aug Shift 1

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

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