Maths ch11 areas related to circles
37 marks. Read the notes, then tap Start test. Answers and your score appear only after you submit.
Quick notes
A sector is the region between an arc and two radii. A segment is the region between a chord and an arc.
For a sector of angle θ in a circle of radius r:
Area of sector = (θ/360°) × πr²
Length of arc = (θ/360°) × 2πr
Area of minor segment = area of sector − area of the triangle formed by the two radii and the chord.
Area of major sector = πr² − area of minor sector. Area of major segment = πr² − area of minor segment.
Use π = 22/7 or 3.14 as the question says.
Part A: Multiple choice 16 questions
1. The area of a sector of angle θ in a circle of radius r is
- (θ/360) × πr²
- (θ/360) × 2πr
- (θ/180) × πr²
- θ × r²
It is the fraction θ/360 of the whole circle's area.
2. The length of an arc of angle θ is
- (θ/360) × πr²
- (θ/360) × 2πr
- θr²
- (θ/180) × πr²
It is the fraction θ/360 of the circumference.
3. The area of a quadrant of a circle of radius 14 cm is
- 616 cm²
- 308 cm²
- 154 cm²
- 44 cm²
(1/4) × (22/7) × 196 = 154 cm².
4. The area of a sector of radius 6 cm and angle 60° (π = 22/7) is
- 132 cm²
- 66/7 cm²
- 36 cm²
- 132/7 cm²
(1/6) × (22/7) × 36 = 132/7 cm².
5. A 14 cm minute hand sweeps an area in 5 minutes of
- 154/3 cm²
- 154 cm²
- 77 cm²
- 616/3 cm²
5 minutes = 30°, so (1/12) × 616 = 154/3 cm².
6. The length of an arc of radius 21 cm and angle 60° is
- 44 cm
- 22 cm
- 66 cm
- 11 cm
(1/6) × 2 × (22/7) × 21 = 22 cm.
7. Area of a minor segment =
- sector + triangle
- circle − sector
- sector − triangle
- triangle − sector
Remove the triangle from the sector.
8. Area of a major sector =
- πr² + minor sector
- 2πr − minor sector
- minor sector − πr²
- πr² − minor sector
The two sectors make the whole circle.
9. If the minor sector angle is 90°, the major sector angle is
- 270°
- 90°
- 180°
- 360°
360° − 90° = 270°.
10. The area of a semicircle of radius 7 cm is
- 154 cm²
- 77 cm²
- 44 cm²
- 22 cm²
(1/2) × (22/7) × 49 = 77 cm².
11. A sector has one quarter of the circle's area when its angle is
- 45°
- 60°
- 90°
- 120°
90/360 = 1/4.
12. An arc of a circle of radius 21 cm is 44 cm long. Its angle is
- 60°
- 90°
- 150°
- 120°
(θ/360) × 132 = 44 gives θ = 120°.
13. The area of a sector of radius 10 cm and angle 90° (π = 3.14) is
- 78.5 cm²
- 314 cm²
- 157 cm²
- 31.4 cm²
(1/4) × 3.14 × 100 = 78.5 cm².
14. A chord of a circle of radius 10 cm subtends 90° at the centre. The minor segment's area (π = 3.14) is
- 50 cm²
- 28.5 cm²
- 78.5 cm²
- 21.5 cm²
78.5 − (1/2 × 10 × 10) = 28.5 cm².
15. An umbrella has 8 equally spaced ribs and radius 45 cm. The area between two ribs (π = 22/7) is
- 22275/7 cm²
- 2025 cm²
- 22275/28 cm²
- 795 cm²
(1/8) × (22/7) × 2025 = 22275/28 cm².
16. A sector with angle 360° has area
- 2πr
- πr²/2
- πr
- πr²
It is the whole circle.
Part B: Fill in the blanks 8 blanks
Part C: Questions and answers 9 questions
25. Find the area of a sector of a circle of radius 6 cm and angle 60°.
26. Find the area of a quadrant of a circle whose circumference is 22 cm.
Area = (1/4) × (22/7) × 12.25 = 77/8 cm² (9.625 cm²).
27. The minute hand of a clock is 14 cm long. Find the area swept in 5 minutes.
28. A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the areas of the minor and major segments (π = 3.14).
Minor segment = 28.5 cm². Major segment = 314 − 28.5 = 285.5 cm².
29. In a circle of radius 21 cm, an arc subtends 60° at the centre. Find the arc length, the sector area and the segment area.
The triangle is equilateral: area = (√3/4) × 441.
Segment = 231 − 441√3/4 cm².
30. A chord of a circle of radius 15 cm subtends 60° at the centre. Find the minor and major segment areas (π = 3.14, √3 = 1.73).
Minor = 20.4375 cm². Major = 706.5 − 20.4375 = 686.0625 cm².
31. A horse is tied with a 5 m rope to a peg at a corner of a 15 m square field. Find the area it can graze (π = 3.14).
Area = (1/4) × 3.14 × 25 = 19.625 m².
32. A car has two wipers that do not overlap. Each blade is 25 cm long and sweeps 115°. Find the total area cleaned (π = 22/7).
33. A lighthouse spreads a red light over a sector of 80° to a distance of 16.5 km. Find the sea area warned (π = 3.14).