These eight questions follow the lessons in circular measure, from converting angles to a problem with mixed units. Write each answer on paper first, then open the answer.
Questions
1. Convert 135° to radians in terms of π.
Answer
135 × π ÷ 180 = 3π/4. As a decimal it is 2.356, between π/2 and π, which fits an angle between 90° and 180°.
2. Convert 2.4 radians to degrees, correct to 1 decimal place.
Answer
2.4 × 180 ÷ π = 137.5, so 137.5°. Size check: 2.4 is a little more than 2 radians (114.6°), so a value near 137° is sensible.
3. A sector has radius 7 cm and angle 0.9 radians. Find its arc length and area.
Answer
Arc length: 7 × 0.9 = 6.3 cm. Area: ½ × 49 × 0.9 = 22.05 cm².
4. A sector has radius 12 cm and arc length 18 cm. Find the angle and the area.
Answer
θ = 18 ÷ 12 = 1.5 radians. Area: ½ × 144 × 1.5 = 108 cm². Check with A = ½rs = ½ × 12 × 18 = 108.
5. The perimeter of a sector is 28 cm and its radius is 8 cm. Find the angle and the area.
Answer
Two radii use 16 cm, so the arc is 28 − 16 = 12 cm. Then θ = 12 ÷ 8 = 1.5 radians.
Area: ½ × 64 × 1.5 = 48 cm². A common slip is to subtract only one radius.
6. A sector has radius 6 cm and angle 2 radians. Find the area of the segment cut off by the chord, to 2 decimal places.
Answer
Area = ½r²(θ − sinθ) = 18 × (2 − sin 2). With sin 2 = 0.9093, the bracket is 1.0907.
Area = 18 × 1.0907 = 19.63 cm². The sector alone is 36 cm², so a segment of 19.63 cm² is plausible.
7. Two sectors share a centre and angle π/4. Their radii are 10 cm and 14 cm. Find the area between them.
Answer
Area = ½(14² − 10²) × π/4 = ½ × 96 × 0.7854 = 48 × 0.7854 = 37.70 cm².
8. A sector has radius 20 cm and angle 150°. Find its area in m², correct to 3 significant figures.
Answer
150° = 5π/6 = 2.618 radians. Area = ½ × 400 × 2.618 = 523.6 cm².
Convert: 523.6 ÷ 10 000 = 0.0524 m² (3 significant figures). Dividing by 100 is the usual wrong step.
If you got these wrong
- Questions 1, 2: revisit converting degrees and radians.
- Questions 3, 4, 5: read arc length and sector area.
- Questions 6, 7: read shaded regions involving sectors.
- Question 8: read applications with consistent units.
Use the timed practice builder to keep practising, and a mistake log to track slips. A teacher can go through your working in online one-to-one Additional Mathematics tuition.