CUET UG 2025 — General Test Geometry

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

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

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

Therefore: $h = d = 2r$h=d=2r

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The volume formulas are:

Cylinder: $V = \pi r^2 h$V=πr2h

Cone: $V = \frac{1}{3}\pi r^2 h$V=31​πr2h

Sphere: $V = \frac{4}{3}\pi r^3$V=34​πr3

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Substituting $h = 2r$h=2r into each formula:

Cylinder:

$$V_1 = \pi r^2 h$$V1​=πr2h

$$V_1 = \pi r^2(2r)$$V1​=πr2(2r)

$$V_1 = 2\pi r^3$$V1​=2πr3

Cone:

$$V_2 = \frac{1}{3}\pi r^2 h$$V2​=31​πr2h

$$V_2 = \frac{1}{3}\pi r^2(2r)$$V2​=31​πr2(2r)

$$V_2 = \frac{2\pi r^3}{3}$$V2​=32πr3​

Sphere:

$$V_3 = \frac{4}{3}\pi r^3$$V3​=34​πr3

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The ratio is:

Cylinder : Cone : Sphere $= 2\pi r^3 : \frac{2\pi r^3}{3} : \frac{4\pi r^3}{3}$=2πr3:32πr3​:34πr3​

Removing the common factor $\pi r^3$πr3:

$$= 2 : \frac{2}{3} : \frac{4}{3}$$=2:32​:34​

Multiplying by 3:

$$= 6 : 2 : 4$$=6:2:4

Dividing by 2:

$$= 3 : 1 : 2$$=3:1:2

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