SSC CGL 2025 Tier-I · 16 September 2025, Shift 1 · Quantitative Aptitude · Question 15 of 100

We transcribed this text from the pictures on a candidate's response sheet for this shift; the answer is the key SSC colour-marked on that sheet. Solution last reviewed 13 September 2026.

Where this paper and its answers come from

SSC does not publish CGL question papers or a public answer key. From 2025 the candidate response sheet prints every question and option as a picture, so the text here is transcribed from those pictures for this shift and checked against them independently; where the layout carries meaning, the picture is shown as well. Each answer is the key SSC colour-marked on that sheet. SSC released a tentative key for challenge and later its final one, both only to candidates behind a login, and the sheet does not say which of the two it carries. The solutions are written by UnlockIAS.

A hemisphere and a cone share the same base and have equal volumes. Given that their common radius is R, determine the height of the cone.

(a)2R
(b)R
(c)5R
(d)4R

Answer

Why

Correct — A. Set the two volume formulas equal, using the same radius R.

Hemisphere volume: 2⁄3 πR³
Cone volume: 1⁄3 πR²h
Set equal: 1⁄3 πR²h = 2⁄3 πR³
Divide both sides by 1⁄3 πR²: h = 2R → option (a)

Why the others are wrong

(b)
R — h = R gives a cone of 1⁄3 πR³, half the hemisphere's 2⁄3 πR³. The cone must be taller than its radius to match.
(c)
5R — h = 5R gives a cone of 5⁄3 πR³, which is 2.5 × the hemisphere's 2⁄3 πR³. Equal volumes need 1⁄3 πR²h = 2⁄3 πR³, and that fixes h at 2R.
(d)
4R — h = 4R gives 4⁄3 πR³, the volume of a full sphere of radius R and double the hemisphere. It fits a cone matched to a whole sphere, not a hemisphere.

Concept

Compare both solids with a cylinder on the same base. A cone of height h is 1⁄3 of the cylinder πR²h. A hemisphere is 2⁄3 of the cylinder of height R, that is 2⁄3 πR³.

With the base shared, πR² cancels, and equal volumes leave 1⁄3 h = 2⁄3 R, so h = 2R.

As a ratio. The cone's height to its radius is 2R : R = 2 : 1, whatever the value of R.

Key facts

Volume of a cone = 1⁄3 πr²h.
Volume of a hemisphere = 2⁄3 πr³.
A cone of height 4r has the volume of a full sphere of radius r (4⁄3 πr³).

Study next

Cone, hemisphere and cylinder of equal radius and height r: volume ratio 1 : 2 : 3Recasting a hemisphere or sphere into conesTotal surface area of a cone mounted on a hemisphere

Common traps

Using the full-sphere volume 4⁄3 πR³ for the hemisphere, which doubles the height to 4R.
Using the cylinder formula πR²h for the cone, which gives h = 2⁄3 R.

Also asked 13 Sep 2025, 16:00, Quant Q.20: a cone and a hemisphere with the same volume and base radius, framed as the ratio of the cone's height to its radius — 2 : 1.

Related PYQs

No directly related past PYQ was found.

Topic: Right Prism, Right Circular Cone, Right Circular Cylinder, Sphere, Hemispheres, Rectangular Parallelepiped, Regular Right Pyramid with triangular or square base — all 99 SSC CGL questions on it, every solved shift.