Measures and solids · Ages 14–15

Arc lengths & sectors worksheet

Covers: Arc length, Area of a sector, Perimeter of a sector, Segments and composite figures, Liu Hui's method for π

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Measures and solids · Ages 14–15

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Name:Date:Score:／48

Write each answer in the space. For word problems, show your working and write a sentence answer.

1.The sector in the picture has a radius of 7 cm and an arc of 7π cm. What is its angle at the centre, in degrees?
Answer:
2.Sector P has a radius of 11 cm and an angle of 180°. Sector Q has a radius of 9 cm and the same arc length as P. What is Q's angle, in degrees?
Answer:
3.The sector in the picture has a radius of 3 cm and an area of 3π cm². What is its angle at the centre, in degrees?
Answer:
4.The sector in the picture has an angle of 72° and an area of 20π cm². What is its radius, in cm?
Answer:
5.The sector in the picture has a radius of 5 cm and an angle of 80°. Correct to 1 decimal place, what is its area, in cm²?
Answer:
6.The sector in the picture has a radius of 16 cm and an angle of 45°. Its arc is π cm, and its perimeter (π + ) cm.
Fill in each blank
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7.The square in the picture has sides of 20 cm; the quarter circle has its centre at one corner and the side as its radius. Taking π as 3.14, what is the shaded area, in cm²?
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8.A semicircle has been cut out of the top side of the square in the picture, whose sides are 12 cm. Taking π as 3.14, what is the shaded part's perimeter, in cm?
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9.The circle in the picture has a radius of 15 cm, and inside it is a regular hexagon with all six corners on the circle. What is the hexagon's perimeter, in cm?
Answer:

10.A sector of radius 3 cm and angle 240° has an arc length of 8π cm.
True or false?
Answer:
11.Taking π as 3.14, a sector of radius 20 cm and angle 180° has a perimeter of 102.8 cm.
True or false?
Answer:
12.A regular 6-gon's perimeter ÷ the circle's diameter gives 3.4641 as an estimate of π. π is more than 3.4641.
True or false?
Answer:

13.A wheel of radius 34 cm rolls along the ground without slipping and moves 170π cm. Find how many degrees it turns: put the steps in order.
- A. Angle = 52 × 360° = 900°
- B. Turns = 170π ÷ 68π = 52
- C. Circumference = 2π × 34 = 68π
Write the letters in the right order
Working:
Answer:
14.A folding fan has ribs 22 cm long and opens out into a sector with an angle of 90°. The arc along its edge is π cm in terms of π, and cm taking π as 3.14.
Fill in each blank
Working:
Answer:
15.A pizza of radius 15 cm is cut into 9 equal slices. Each slice has an angle of ° at the centre and an area of π cm².
Fill in each blank
Working:
Answer:
16.A wire 54 cm long is bent exactly into the edge of a sector of radius 19 cm. To the nearest degree, what is the sector's angle at the centre?
Working:
Answer:
17.A car's wiper reaches 46 cm from its pivot, the rubber blade from 18 cm to 46 cm out, and it swings through 135°, wiping a ring sector. Its area is π cm² in terms of π, and cm² taking π as 3.14.
Fill in each blank
Working:
Answer:
18.A hexagonal gazebo stands inside a circular path of radius 17 m, its six corners just on the path. The gazebo's perimeter is m; taking π as 3.14, the path is m round, m more than the gazebo.
Fill in each blank
Working:
Answer:

Show the answers and working

Arc lengths & sectors worksheet: Answers and working

1. 180°
θ360 × 2π × 7 = 7π, so θ = 7 × 180 ÷ 7 = 180°. Or: what share of the circumference 2π × 7 the arc is, times 360°.
2. 220°
P's arc is 180360 × 2π × 11 = 11π cm; for Q, θ360 × 2π × 9 = 11π, so θ = 11 × 180 ÷ 9 = 220°. Radius and angle go inversely: 11 × 180 = 9 × 220.
3. 120°
θ360 × π × 3² = 3π, so θ = 3 × 360 ÷ 3² = 120°.
4. 10 cm
72360 × πr² = 20π, so r² = 20 × 360 ÷ 72 = 100 and r = 10 cm.
5. 17.5 cm²
80360 × π × 5² with the calculator's π, to 1 decimal place: about 17.5 cm².
6. 4, 4, 32
Arc 45360 × 2π × 16 = 4π; the perimeter adds two radii, 2 × 16 = 32.
7. 86 cm²
Shaded = square − quarter circle = 400 − 14 × 3.14 × 20² = 400 − 314 = 86 cm².
8. 54.84 cm
Three straight sides 3 × 12 = 36, plus the semicircle's arc 3.14 × 6 = 18.84 (the side cut away is no longer on the edge): 54.84 cm.
9. 90 cm
Joining the centre to the corners makes 6 equilateral triangles, so each side equals the radius 15: 6 × 15 = 90 cm.

10. False
Arc = 240360 × 2π × 3 = 4π cm: the circumference is 2πr, not πr nor 4πr.
11. True
Perimeter = arc + two radii = 102.8 cm; the arc alone leaves out 2 × 20.
12. False
π ≈ 3.1416: a polygon inside the circle is shorter, its estimate below π; one outside is longer, above π.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. C → B → A
One turn moves one circumference, 68π cm; 170π is 52 turns, 900°.
14. 11, 34.54
90360 × 2π × 22 = 11π cm; 11 × 3.14 = 34.54 cm.
15. 40, 25
360 ÷ 9 = 40°; area 40360 × π × 15² = π × 15² ÷ 9 = 25π cm².
16. 48°
The two radii take 2 × 19, so the arc = 54 − 2 × 19 = 16 cm; θ = 16 ÷ (2π × 19) × 360 ≈ 48°.
17. 672, 2110.08
135360 × π(46² − 18²) = 672π; × 3.14 = 2110.08 cm².
18. 102, 106.76, 4.76
Each side equals the radius: 6 × 17 = 102; 2 × 3.14 × 17 = 106.76; 106.76 − 102 = 4.76.

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