Arc lengths & sectors · Ages 14–15
Area of a sector
Sector area = θ360 × πr² = 12 lr.
How to do it
Method
Sector area = θ/360 × πr² = 1/2 lr
r = 6, θ = 60°: 36π ÷ 6 = 6π cm²
Another way
From the arc: 1/2 lr
12 × 2π × 6 = 6π
In full
A sector's area is θ360 of the circle's too: sector area = θ360 × πr². Radius 6 cm, angle 60°: the circle is π × 62 = 36π, so the sector is 36π ÷ 6 = 6π cm². Another way: from the arc length l, sector area = 12 × l × r, like a triangle with base l and height r. Arc 2π, radius 6: 12 × 2π × 6 = 6π, the same. The other way round, from the area: θ = sector ÷ circle × 360; the radius: r² = sector area × 360 ÷ (θπ), then the square root. Careful: r² is r × r; halve a diameter first.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
The sector in the picture has a radius of 10 cm and an area of 40π cm². What is its angle at the centre, in degrees?
Show the answer and working
Answer: 144°
θ360 × π × 102 = 40π, so θ = 40 × 360 ÷ 102 = 144°.
Example 2
A sector of radius 6 cm and angle 90° has an area of 18π cm².
True or false?
Show the answer and working
Answer: False
Area = 90360 × π × 62 = 9π cm².
Example 3
The sector in the picture has a radius of 15 cm and an angle of 195°. Correct to 1 decimal place, what is its area, in cm²?
Show the answer and working
Answer: 382.9 cm²
195360 × π × 152 with the calculator's π, to 1 decimal place: about 382.9 cm².
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The sector in the picture has a radius of 12 cm and an angle of 45°. What is its area?
- A. 72π cm²
- B. 3π cm²
- C. 144π cm²
- D. 18π cm²
Choose the right answer
Answer:
2.The sector in the picture has a radius of 20 cm and an angle of 45°. What is its area, in terms of π?
Answer:
3.The sector in the picture has a radius of 6 cm and an arc of 3π cm. What is its area, in terms of π?
Answer:
4.A sector of radius 6 cm and angle 300° has an area of 30π cm².
True or false?
Answer:
5.The sector in the picture has a radius of 10 cm and an arc of 21 cm. Its area is cm², and its angle, to the nearest degree, °.
Fill in each blank
Answer:
All the answers
- 1. D. 18π cm²
- 2. 50 π cm²
- 3. 9 π cm²
- 4. True
- 5. 105, 120
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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