CUET UG 2025 — General Test Geometry

2025mcqmedium

If a cone and sphere have equal radii and volumes, then determine the ratio of the diameter of the sphere to the height of the cone?

Official previous-year question

Held on 3 Jun 2025 · Verified 13 Jul 2026.

Options

A
3:1
B
1:2
C
6:1
D
1:3

Did you get this right?

Solution

A cone and sphere have equal radii $r$ and equal volumes.

Volume of sphere: $V_{sphere} = \frac{4}{3}\pi r^3$

Volume of cone: $V_{cone} = \frac{1}{3}\pi r^2 h$

where $h$ is the height of the cone.

Since the volumes are equal:

$\frac{4}{3}\pi r^3 = \frac{1}{3}\pi r^2 h$

$4\pi r^3 = \pi r^2 h$

$4r^3 = r^2 h$

$4r = h$

The height of the cone is $4r$ .

The diameter of the sphere is $2r$ .

The height of the cone is $4r$ .

Ratio of diameter to height:

$\frac{2r}{4r} = \frac{1}{2}$

Therefore, the ratio of the diameter of the sphere to the height of the cone is $1:2$ .

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CUET UG 2025 — General Test Geometry

2025mcqmedium

If a cone and sphere have equal radii and volumes, then determine the ratio of the diameter of the sphere to the height of the cone?

Official previous-year question

Held on 3 Jun 2025 · Verified 13 Jul 2026.

Options

A
3:1
B
1:2
C
6:1
D
1:3

Did you get this right?

Solution

A cone and sphere have equal radii $r$ and equal volumes.

Volume of sphere: $V_{sphere} = \frac{4}{3}\pi r^3$

Volume of cone: $V_{cone} = \frac{1}{3}\pi r^2 h$

where $h$ is the height of the cone.

Since the volumes are equal:

$\frac{4}{3}\pi r^3 = \frac{1}{3}\pi r^2 h$

$4\pi r^3 = \pi r^2 h$

$4r^3 = r^2 h$

$4r = h$

The height of the cone is $4r$ .

The diameter of the sphere is $2r$ .

The height of the cone is $4r$ .

Ratio of diameter to height:

$\frac{2r}{4r} = \frac{1}{2}$

Therefore, the ratio of the diameter of the sphere to the height of the cone is $1:2$ .

Your note

CUET UG Geometry in other years

Geometry 2024 Geometry 2023 Geometry 2022

Keep practicing Geometry

Work through every CUET UG Geometry PYQ, year by year.

Browse all General Test topics

CUET UG 2025 — General Test Geometry

Options

Did you get this right?

Solution

Your note

CUET UG Geometry in other years

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General Knowledge

Logical Reasoning

Arithmetic

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CUET UG 2025 — General Test Geometry

Options

Did you get this right?

Solution

Your note

CUET UG Geometry in other years

More CUET UG Geometry questions

Other General Test topics for CUET UG

General Knowledge

Logical Reasoning

Arithmetic

Quantitative Reasoning

Current Affairs

Keep practicing Geometry