SSC CGL 2024 Tier-I · 10 September 2024, Shift 3 · Quantitative Aptitude · Question 17 of 100

Text as printed on a candidate's response sheet for this shift; answer per SSC's tentative key as marked on that sheet. Solution last reviewed 6 September 2026.

Where this paper and its answers come from

SSC does not publish CGL question papers or a public answer key. The questions and options here are reproduced as printed on a candidate's own response sheet for this shift — the Commission's text, unchanged. Each answer is SSC's tentative key as marked on that sheet; SSC allowed challenges to it and never publishes a final key. The solutions are written by UnlockIAS.

What is the central angle of a sector of a circle whose area and perimeter are, respectively, equal to 209 cm 2 and 63 cm if it is given that its radius is a non-negative integer (roundup to one decimal place)?

(a)49.5°
(b)55.5°
(c)53.5°
(d)51.5°

Answer

Why

Correct — A. Call the radius r and the arc length l.

Perimeter of a sector = 2r + l = 63
Area of a sector = ½ × r × l = 209, so r l = 418

Substitute l = 63 − 2r:
r(63 − 2r) = 418
2r² − 63r + 418 = 0
r = (63 ± 25) ÷ 4 = 22 or 9.5

The radius is given as an integer, so r = 22 and l = 63 − 44 = 19.

Central angle = l/r radian = 19/22 radian = (19/22) × 180 ÷ π
With π = 22/7 that is 49.46°, with π = 3.14 it is 49.51° — 49.5° either way → option (a)

Why the others are wrong

(b)
55.5° — 55.5° cannot meet both conditions. With a 63 cm perimeter it forces a radius of about 21.2 cm, which is not a whole number, and an area near 218 cm² rather than 209.
(c)
53.5° — 53.5° with the same 63 cm perimeter needs a radius of about 21.5 cm and gives an area near 215 cm². Both the integer requirement and the 209 cm² fail.
(d)
51.5° — 51.5° is the closest wrong option, needing a radius of about 21.7 cm and an area near 212 cm² — which is exactly why the question insists the radius be a whole number.

Concept

A sector has two measurements that both involve its arc. Its perimeter is 2r + l, two radii plus the arc, not 2πr. Its area is ½ r l, which mirrors ½ × base × height.

Given both, you have two equations in r and l. Eliminating l leaves a quadratic in r, and solving that is far quicker than carrying degrees through the algebra.

The angle follows last, from l = rθ with θ in radians, converted at 180/π degrees to the radian.

The quadratic has two positive roots, 22 and 9.5, and both describe a real sector of area 209 cm² and perimeter 63 cm — the second is a reflex sector of about 265°. The clause 'radius is a non-negative integer' is what selects 22.

Key facts

The perimeter of a sector is 2r + l, where l is the arc length.
The area of a sector is ½ × r × l.
Here r = 22 cm and l = 19 cm, so the central angle is 19/22 radian, about 49.5 degrees.

Study next

Arc length and sector area written in the degree form as well as the radian formConverting radians to degrees using π = 22/7 and checking the roundingPerimeter of a segment as against the perimeter of a sector

Common traps

Using 2πr for a sector's perimeter instead of 2r + l
Taking the other root r = 9.5 and reporting the reflex angle of about 265°
Rounding l/r before converting to degrees, which shifts the one-decimal answer

The item supplies area and perimeter and asks for the fourth quantity, the angle.

The clause 'it is given that its radius is a non-negative integer' is doing real work — it exists only to reject one root of the quadratic, so read such conditions as instructions rather than decoration.

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