Grade 10 core mathematics – Area of a Part of a Circle (12 lessons) Quiz
1. Find the area of a sector of a circle with radius 6 cm and central angle 60°.
Sector area = (θ/360)·πr² = (60/360)·π·6² = (1/6)·36π = 6π cm².
2. A semicircular swimming pool has radius 5 m. What is its area?
Area of semicircle = (1/2)·πr² = 0.5·π·5² = 0.5·25π = 12.5π m².
3. Find the area of a quarter circle of radius 8 cm.
Quarter circle area = (1/4)·πr² = (1/4)·π·8² = (1/4)·64π = 16π cm².
4. A circle has radius 10 cm and a central angle of 60°. Find the area of the corresponding minor segment (area between chord and arc).
Segment area = sector area − triangle area. Sector = (60/360)·π·10² = 100π/6 = 50π/3. Triangle = 1/2·r²·sin60° = 0.5·100·(√3/2) =25√3. So area = 50π/3 − 25√3 cm².
5. An arc length is 7π/3 cm on a circle of radius 7 cm. Find the area of the sector formed by this arc.
θ (rad) = s/r = (7π/3)/7 = π/3. Sector area = 1/2·r²·θ = 1/2·49·(π/3) = 49π/6 cm².
6. A sector has area 9π cm² and central angle 90°. What is the radius of the circle?
Sector area = (θ/360)·πr². Here (90/360)=1/4 so (1/4)·πr² = 9π ⇒ r² = 36 ⇒ r = 6 cm.
7. Find the area of the segment of a circle with radius 9 cm and central angle 120° (minor segment).
Sector area = (120/360)·π·9² = (1/3)·81π = 27π. Triangle area = 1/2·r²·sin120° = 0.5·81·(√3/2) = (81√3)/4. Segment = sector − triangle = 27π − (81√3)/4 cm².
8. A roundabout is shaped like a sector with radius 7 m and central angle 135°. What is its area?
Area = (θ/360)·πr² = (135/360)·π·7² = (3/8)·49π = 147π/8 m².
9. Find the area of the shaded region that is a semicircular ring between radii 10 m and 6 m.
Area of semicircular ring = (1/2)·π(R² − r²) = 0.5·π(100 − 36) = 0.5·π·64 = 32π m².
10. A pizza has radius 12 cm. What is the area of a 45° slice?
Sector area = (45/360)·π·12² = (1/8)·π·144 = 18π cm².
11. A chord of length 12 cm is in a circle of radius 10 cm. Find the area of the sector determined by the chord (use degrees, give answer to two decimal places).
Chord c = 2r sin(θ/2) ⇒ sin(θ/2)=12/(2·10)=0.6 ⇒ θ/2≈36.87° ⇒ θ≈73.74°. Sector area = (θ/360)·π·10² ≈ (73.74/360)·π·100 ≈ 64.36 cm².
12. An arc length of a sector is 22 cm and the radius of the circle is 7 cm. Find the area of that sector.
Sector area = (1/2)·r·s since area = 1/2·r²·θ and θ = s/r. So area = 0.5·r·s = 0.5·7·22 = 77/2 = 38.5? Wait compute properly: 0.5·r·s = 0.5·7·22 = 77. However earlier derivation was wrong—re-evaluate: sector area = 1/2 r·s (correct). 0.5·7·22 = 77 cm². But this contradicts expectation. Re-check: using θ = s/r = 22/7·1/7? Actually s=22 cm and r=7 cm ⇒ θ = s/r =22/7 rad ≈3.1429 rad. Sector area = 1/2·r²·θ = 0.5·49·(22/7) = 0.5·49·22/7 = 0.5·7·22 =77 cm². Therefore the correct area is 77 cm².
13. A semicircular garden has diameter 20 m. What is its area?
Radius = 10 m. Semicircle area = (1/2)·π·10² = 50π m².
14. Find the area of the triangle formed by two radii and the included angle of 120° in a circle of radius 13 cm.
Triangle area = 1/2·r²·sinθ = 0.5·13²·sin120° = 0.5·169·(√3/2) = 169·√3/4 = 42.25√3 cm².
15. A circle of radius 12 cm has a sector with central angle 30°. Find the area of the corresponding segment (sector minus triangle).
Sector area = (30/360)·π·12² = 1/12·144π = 12π. Triangle area = 1/2·r²·sin30° = 0.5·144·0.5 = 36. Segment = 12π − 36 cm².
16. Find the area of a quarter circle whose diameter is 14 cm.
Radius = 7 cm. Quarter area = (1/4)·π·7² = (1/4)·49π = 49π/4 cm².
17. A circular window has radius 3 m. What is the area of the top quarter covered by a metal decoration?
Quarter area = (1/4)·π·3² = (1/4)·9π = 9π/4 m².
18. Find the area of a sector with radius 5 cm and central angle 2 radians.
Sector area (radians) = 1/2·r²·θ = 0.5·25·2 = 25 cm².
19. Find the arc length subtended by a 120° angle in a circle of radius 9 cm.
Arc length = (θ/360)·2πr = (120/360)·2π·9 = (1/3)·18π = 6π cm.
20. An annular sector has outer radius 15 cm, inner radius 9 cm and central angle 60°. Find its area.
Area = (θ/360)·π(R² − r²) = (60/360)·π(225 − 81) = (1/6)·π·144 = 24π cm².
21. A sector has area 18π cm² and radius 6 cm. What is the central angle?
Sector area = (θ/360)·πr². So 18π = (θ/360)·π·36 ⇒ θ/360 = 18/36 = 1/2 ⇒ θ = 180°.
22. A chord of a circle of radius 20 cm has length 20 cm. Find the area of the corresponding minor segment.
sin(θ/2)=c/(2r)=20/40=0.5 ⇒ θ/2=30° ⇒ θ=60°. Sector area = (60/360)·π·400 = (1/6)·400π = 200π/3. Triangle area = 1/2·r²·sin60° = 0.5·400·(√3/2) = 100√3. Segment = 200π/3 − 100√3 cm².
23. A circular garden radius 9 m has a path of width 2 m around it. What is the area of the path in exactly one quarter of the circle?
Outer radius = 11, inner = 9. Full annulus area = π(11² − 9²) = π(121 − 81) = 40π. Quarter = 40π/4 = 10π. Wait re-evaluate: inner radius 9 outer 11 ⇒ area difference = π(121−81)=40π ⇒ quarter =10π. The correct choice given is 8π, which is incorrect. Correction needed: quarter area is 10π m². (Note: the correct answer should be 10π m².)
24. An arc of length 11 cm corresponds to a sector whose area is 38.5 cm². Find the radius of the circle.
Sector area = 1/2·r·s. So r = 2·area/s = 2·38.5/11 = 77/11 = 7 cm.