The angle (in degrees) made by a sector having area one-sixth of the area of a semicircle is
- (a)15°
- (b)30°
- (c)45°
- (d)60°
Correct — B, 30°. Sector area is proportional to the angle at the centre, so the whole comparison can be made in degrees. A semicircle is itself a sector of 180°, with area πr²/2. One-sixth of that is πr²/12. Setting (θ/360) × πr² = πr²/12 cancels the radius entirely and gives θ/360 = 1/12, so θ = 30°. The shorter route is to notice that taking one-sixth of a shape whose angle is 180° means taking one-sixth of the angle: 180 ÷ 6 = 30.
- (a)15° — 15° is one-sixth of 90°, which would be the answer if the comparison were with a quadrant. The question compares with a semicircle, whose angle is 180°.
- (c)45° — 45° is one-eighth of the full circle and one-quarter of the semicircle. Its area would be πr²/8, not πr²/12.
- (d)60° — 60° is one-sixth of the full circle of 360°, which is the answer to a different question. One-sixth of the semicircle is half of that.
For a circle of radius r, a sector subtending θ at the centre has area (θ/360) × πr² and arc length (θ/360) × 2πr. Both scale linearly with θ, so any question that compares two sectors of the same circle can be answered in angles alone, without ever writing π or r.
Two words decide this item: semicircle and one-sixth. A candidate who reads 'circle' where the paper says 'semicircle' lands on 60°, which is offered as option (d) for exactly that reason. Writing the fraction of the full circle keeps the reading honest: the target area is (1/6) × (1/2) = 1/12 of the disc, and 1/12 of 360° is 30°.
- Sector area = (θ/360) × πr², so area and central angle are directly proportional.
- A semicircle is a sector of 180°; a quadrant is a sector of 90°.
- One-sixth of a 180° sector is a 30° sector, and 30° is one-twelfth of the full circle.
- The arc length of a 30° sector of radius r is πr/6.
- Area of a segment = area of the sector minus the area of the triangle formed by the two radii and the chord.
Because area is proportional to angle, the radius cancels and the whole item reduces to dividing 180 by 6.
- Comparing with the full circle instead of the semicircle and answering 60°.
- Reaching for the area formula and getting lost in π and r when only the angle is needed.
- Confusing one-sixth of the area with one-sixth of the arc — they happen to agree here, but not for segments.
A one-step mensuration item where the marks turn on reading the reference shape correctly, not on the formula.
A solid spherical ball made of iron is melted and two new balls are made whose diameters are in the ratio of 1 : 2. The ratio of the volume of the smaller new ball to the original ball is
- (a) 1 : 3
- (b) 1 : 5
- (c) 2 : 9
- (d) 1 : 9
Answer(d) 1 : 9
Mensuration set up as a proportion rather than a calculation. There the volume scales as the cube of the diameter; here the sector area scales directly with the central angle, and in both cases the ratio does all the work.
- practice — not a real PYQ
The area of a sector of angle 60° in a circle of radius 6 cm is
- (a)6π cm²
- (b)9π cm²
- (c)12π cm²
- (d)18π cm²
Answer(a) 6π cm² — (60/360) × π × 36 = 6π.
- practice — not a real PYQ
A sector has area one-third of the area of a quadrant of the same circle. Its central angle is
- (a)15°
- (b)22.5°
- (c)30°
- (d)45°
Answer(c) 30° — a quadrant subtends 90°, and one-third of 90° is 30°.