Geometry

Solids

Hard

Consider the following statements, which of them is/are correct?

A. If the height of cylinder is doubled, the area of curved surface is doubled.

B. If the radius of a hemispherical solid is doubled, its total surface area becomes fourfold.

C. If a hemisphere and cone have equal bases and equal heights, then the ratio of curved surface area is $\sqrt{2}:1$ .

B and C only

A and C only

A and B only

A, B and C

✅ Correct Option: 4

Statement A: Cylinder Height Doubled

Property	Formula	Value
Original CSA	$2\pi rh$	$2\pi rh$
New CSA (height = $2h$ )	$2\pi r(2h)$	$4\pi rh$
Ratio	New ÷ Original	$2$

Conclusion: CSA doubles. ✓ Correct

Statement B: Hemisphere Radius Doubled

Property	Formula	Value
Original TSA	$3\pi r^2$	$3\pi r^2$
New TSA (radius = $2r$ )	$3\pi (2r)^2$	$12\pi r^2$
Ratio	New ÷ Original	$4$

Conclusion: TSA becomes fourfold. ✓ Correct

Statement C: Hemisphere vs Cone (Equal Base & Height)

Given: Base radius = $r$ , Height = $r$ (hemisphere height equals its radius)

Shape	CSA Formula	Calculation	Result
Hemisphere	$2\pi r^2$	$2\pi r^2$	$2\pi r^2$
Cone	$\pi r l$	$\pi r \cdot r\sqrt{2}$	$\pi r^2\sqrt{2}$

Note: Slant height $l = \sqrt{r^2 + r^2} = r\sqrt{2}$

$\dfrac{\text{Hemisphere CSA}}{\text{Cone CSA}} = \dfrac{2\pi r^2}{\pi r^2\sqrt{2}} = \dfrac{2}{\sqrt{2}} = \sqrt{2} : 1$

Hence, the ratio is $\sqrt{2} : 1$ . ✓ Correct

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Solution

Statement A: Cylinder Height Doubled

Statement B: Hemisphere Radius Doubled

Statement C: Hemisphere vs Cone (Equal Base & Height)

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