SSC CGL 2025 Tier-I · 12 September 2025, Shift 2 · Quantitative Aptitude · Question 19 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.

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What is the central angle of a sector with an arc length of 10 cm in a circle of radius 5 cm?

(a)6 radians
(b)3 radians
(c)2 radians
(d)4 radians

Answer

Why

Correct — C. In radians, a central angle is arc length ÷ radius, because arc = radius × angle.

θ = l ÷ r
θ = 10 ÷ 5 = 2
Check: 5 × 2 = 10 cm, the given arc ✓

2 radians → option (c)

Why the others are wrong

(a)
6 radians — 6 radians would need an arc of 5 × 6 = 30 cm. That is nearly the whole circumference, 2π × 5 ≈ 31.4 cm, not a 10 cm arc.
(b)
3 radians — 3 radians would need an arc of 5 × 3 = 15 cm, half as long again as the 10 cm arc given.
(d)
4 radians — 4 radians would need an arc of 5 × 4 = 20 cm, twice the given 10 cm. Arc ÷ radius is 2, not 4.

Concept

A radian is the central angle whose arc equals the radius. An arc of r subtends 1 radian, an arc of 2r subtends 2 radians.

That gives the working rule l = rθ, with θ in radians, so θ = l ÷ r. Both lengths are in cm and cancel, which is why a radian carries no unit.

Here the arc is exactly twice the radius, so the angle is 2 radians, about 114.6°.

l = rθ needs θ in radians. With θ in degrees the arc is (θ ⁄ 360) × 2πr, and 114.6° gives (114.6 ⁄ 360) × 2π × 5 ≈ 10 cm, the same arc.

Key facts

Arc length l = rθ, with θ in radians.
1 radian = 180° ⁄ π ≈ 57.3°, so 2 radians ≈ 114.6°.
A full turn is 2π ≈ 6.28 radians, so 6 radians is just short of one turn.
Sector area = ½ r²θ = ½ × l × r.

Study next

Converting between radians and degreesSector area from arc length and radius: A = ½ × l × rArc length with the angle in degrees: (θ ⁄ 360) × 2πr

Common traps

Using l = rθ with θ in degrees. The formula needs radians, so convert first or use (θ ⁄ 360) × 2πr.
Reading 2 radians as 2°. A radian is about 57.3°, so this sector's angle is about 114.6°, an obtuse one.

Arc and radius together also decide 9 Sep 2024, 12:30, Quant Q.19, where the sector area is ½ × l × r = ½ × 11 × 15.75 = 86.625 cm².

Radian-to-degree conversion appears at Quant Q.17 of this paper (5π⁄4 = 225°) and on 14 Sep 2025, 16:00, Quant Q.18 (2.5 radians ≈ 143.24°).

Related PYQs

No directly related past PYQ was found.

Topic: Circle and its chords, tangents, angles subtended by chords of a circle, common tangents to two or more circles — all 140 SSC CGL questions on it, every solved shift.