This set has twelve original questions, ordered from easier to harder, covering all five lessons in radians, arcs and sectors. Questions 1 to 3 are conversions, 4 to 9 use arc length, sector area and perimeter, and 10 to 12 mix skills and need more steps.
Attempt each question on paper and write your working as you would in an exam. Set the calculator to radian mode first, and test it with sin 1 = 0.8415. Only then open the answer, mark the ones you got wrong and use the routing list at the end.
Questions
1. Convert 150° to radians, in terms of π.
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150 × π/180 = 150π/180. Divide top and bottom by 30 to get 5π/6.
5π/6 rad. Check: 150° is 30° short of 180°, and π − π/6 = 5π/6. ✓
2. Convert 11π/12 rad to degrees.
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11π/12 × 180/π = 11 × 180/12 = 11 × 15 = 165.
165°. Check: it is just under 180°, which is π. ✓
3. Convert 2.5 rad to degrees, to 1 decimal place.
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2.5 × 180/π = 450/π = 143.239…
143.2°. Check: 2.5 × 57.3° is about 143°. ✓
4. A sector has radius 7.5 cm and angle 0.8 rad. Find the arc length.
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s = rθ = 7.5 × 0.8 = 6.
6 cm
5. The minute hand of a clock is 14 cm long. Find the distance moved by its tip in 20 minutes, in terms of π and to 3 significant figures.
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In 20 minutes the hand turns 20/60 = 1/3 of a full turn, so θ = 2π/3. Then s = 14 × 2π/3 = 28π/3 = 29.321…
28π/3 cm, which is 29.3 cm. Check: one third of the circumference 28π is 28π/3. ✓
6. A sector has radius 15 cm and angle 72°. Find the arc length in terms of π and to 3 significant figures.
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72° = 72π/180 = 2π/5 rad. So s = 15 × 2π/5 = 6π = 18.849…
6π cm, which is 18.8 cm. Check: 72/360 = 1/5, and one fifth of 30π is 6π. ✓
7. A sector has radius 12 cm and angle 5π/6. Find its area in terms of π and to 3 significant figures.
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A = ½ × 12² × 5π/6 = 72 × 5π/6 = 60π = 188.495…
60π cm², which is 188 cm². Check: 5π/6 is 150°, and 150/360 × 144π = 60π. ✓
8. A sector has area 27 cm² and angle 1.5 rad. Find the radius, the arc length and the perimeter.
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27 = ½ r² × 1.5 = 0.75r², so r² = 36 and r = 6. Arc length s = rθ = 6 × 1.5 = 9. Perimeter = 2r + s = 12 + 9 = 21.
r = 6 cm, arc = 9 cm, perimeter = 21 cm. Check: A = ½ r s = ½ × 6 × 9 = 27. ✓
9. A sector has radius 9 cm and perimeter 33 cm. Find the angle in radians and the area of the sector.
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Arc = 33 − 2 × 9 = 15. θ = 15/9 = 5/3. Area = ½ r s = ½ × 9 × 15 = 67.5.
θ = 5/3 rad (1.67), area = 67.5 cm². Check: ½ × 81 × 5/3 = 67.5. ✓
10. A sector has radius 12 cm and angle 2π/3. Find the exact area of the segment cut off by the chord, then give it to 3 significant figures.
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Sector: ½ × 144 × 2π/3 = 48π. Triangle: ½ × 144 × sin(2π/3) = 72 × √3/2 = 36√3. Segment = 48π − 36√3 = 150.796 − 62.354 = 88.443…
48π − 36√3 cm², which is 88.4 cm². Check: the segment is smaller than the sector 150.8 cm², and larger than zero. ✓
11. A sector OAB has radius 10 cm and angle 1.6 rad. Find the perimeter of the segment (the arc AB plus the chord AB) and the area of the segment, to 3 significant figures.
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Arc = 10 × 1.6 = 16. Chord = 2r sin(θ/2) = 20 × sin 0.8 = 20 × 0.71736 = 14.347. Perimeter = 16 + 14.347 = 30.347.
Area = ½ r² (θ − sin θ) = 50 × (1.6 − 0.99957) = 50 × 0.60043 = 30.021.
Perimeter 30.3 cm, area 30.0 cm². Check: the chord (14.3) is shorter than the arc (16). ✓
12. A sector has perimeter 28 cm and area 48 cm². Find the two possible radii and the matching angles.
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Arc = 28 − 2r, so ½ r (28 − 2r) = 48, giving r(14 − r) = 48, so r² − 14r + 48 = 0. Factorising: (r − 6)(r − 8) = 0, so r = 6 or r = 8.
If r = 6, arc = 28 − 12 = 16 and θ = 16/6 = 8/3. If r = 8, arc = 28 − 16 = 12 and θ = 12/8 = 3/2.
r = 6 cm with θ = 8/3 rad (2.67), or r = 8 cm with θ = 3/2 rad. Check areas: ½ × 6 × 16 = 48 and ½ × 8 × 12 = 48. Both angles are below 2π. ✓
If you got these wrong
- Conversion answers off by a factor of about 57 or by an upside-down fraction (questions 1 to 3): go to converting degrees and radians precisely.
- Arc lengths that are far too large, or a degree angle used in s = rθ (questions 4 to 6): go to finding an arc length using radian measure.
- Sector areas that are twice too big, or a missing half (questions 7 and 8): go to calculating a sector area.
- Segment values that are nearly as large as the sector, or a calculator in degree mode (questions 10 and 11): go to comparing a sector with an enclosed triangle.
- Perimeters that leave out the two radii, or a quadratic with one root dropped (questions 9 and 12): go to solving a perimeter condition involving an arc.
Record each slip in the mistake log and retest queue, and retry a similar question in a few days. The non-calculator working trainer and the quadratic structure explorer support the exact arithmetic and the quadratic step.
If one type of error stays after two rounds of revision, a teacher can look at your written solutions in online one-to-one Additional Mathematics tuition.