Sector area is a common GCSE Maths topic tested across AQA, Edexcel, and OCR on both Foundation and Higher papers. A sector is a "slice" of a circle, like a pizza slice, and its area depends on the radius and the angle at the centre. This guide covers the area formula, perimeter of a sector, and how to handle reverse problems where you find the angle or radius.
What Is Sector Area?
A sector is the region enclosed by two radii and an arc. The area of a sector is a fraction of the full circle's area. That fraction is determined by the angle at the centre divided by 360°.
Key Formulas
Step-by-Step Method
- Identify the radius (r) and the angle at the centre (θ).
- Write the formula: Area = (θ / 360) × πr².
- Substitute the values and calculate.
- For the perimeter, add the arc length to twice the radius (the two straight edges).
- Round to the required accuracy or leave in terms of π.
Worked Example 1 — Foundation Level
Question: Find the area of a sector with radius 6 cm and angle 90°. Give your answer to 1 decimal place.
Working:
Area = (90 / 360) × π × 6²
= (1/4) × 36π
= 9π
= 28.274...
Answer: Area = 28.3 cm² (1 d.p.)
Worked Example 2 — Higher Level
Question: A sector has a radius of 10 cm and an area of 50π cm². Find the angle at the centre.
Working:
Area = (θ / 360) × πr²
50π = (θ / 360) × π × 10²
50π = (θ / 360) × 100π
Divide both sides by π: 50 = (θ / 360) × 100
θ / 360 = 50 / 100 = 1/2
θ = 180°
Answer: The angle at the centre is 180°.
Worked Example 3 — Exam Style
Question: A sector has a radius of 8 cm and an angle of 135°. Find the perimeter of the sector. Give your answer to 1 decimal place.
Working:
Arc length = (135 / 360) × 2 × π × 8 = (3/8) × 16π = 6π = 18.849...
Perimeter = 2 × 8 + arc length = 16 + 18.849... = 34.849...
Answer: Perimeter = 34.8 cm (1 d.p.)
Common Mistakes
- Confusing arc length and sector area formulas. Arc length uses 2πr (circumference); sector area uses πr² (area). Make sure you pick the right one.
- Forgetting the two radii in the perimeter. The perimeter of a sector includes the arc and the two straight edges (radii), so you add 2r to the arc length.
- Squaring the diameter instead of the radius. If the question gives the diameter, halve it before squaring.
Exam Tips
- When asked for perimeter of a sector, many students only calculate the arc length and forget the two radii. Always add 2r.
- Show your substitution into the formula clearly — this earns method marks.
- If the question says "in terms of π", simplify the numerical coefficient and leave π in your answer.
- Sector area and arc length often appear together. Read the question carefully to determine which one is asked for.
Practice Questions
Q1 (Foundation): Find the area of a sector with radius 7 cm and angle 60°. Give your answer to 1 d.p.
Q2 (Foundation): A semicircular sector has a diameter of 20 cm. Find its area in terms of π.
Q3 (Higher): A sector has an area of 75 cm² and a radius of 10 cm. Find the angle at the centre to 1 d.p.
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Related Topics
Summary
- The area of a sector is (θ / 360) × πr², where θ is the angle at the centre and r is the radius. The perimeter of a sector is the arc length plus twice the radius. You can rearrange the formula to find the angle or radius when the area is given. Always check whether the question gives radius or diameter, and remember to include both radii when calculating the perimeter.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
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