Treat a sector as a fraction 360θ of the whole circle to find arc length, sector area, perimeter or the angle. It is on Foundation and Higher, aimed at grade 5. Free questions with answers are on this page. Worked examples show each line before you try the questions.
A sector is a pizza slice of a circle. The fraction of the circle is 360θ. Arc length =360θ×2πr. Sector area =360θ×πr2. A segment (the crust cap) is sector minus the triangular slice from the centre.
Perimeter of a sector is arc plus two radii. Do not use the chord as if it were the arc. Exact answers keep π; calculator answers round at the end. If θ is missing, rearrange the arc or area formula.
A sector is a fraction 360θ of the circle. Arc =360θ×2πr, area =360θ×πr2. A quarter is θ=90; a semicircle is 180. Add radii for the perimeter of a sector.
Objectives
Find the area of a sector in terms of π.
Find an arc length, and the fraction of the circle.
Find the perimeter of a sector, including the two radii.
Key points
Find an arc length as a fraction of the circumference 2πr.
Find a sector area as a fraction of the circle area πr2.
Find the perimeter of a sector by adding two radii to the arc length.
Work backwards to find the angle (or radius) from a given arc length or area.
Worked examples
Example 1
Work out the area of a sector with angle 60 degrees and radius 6. Give your answer in terms of π.
2 marks
Find the fraction of the circle: 60 / 360 = 1/6
Calculate full circle area: π × 6^2 = 36π
Multiply fraction by area: (1/6) × 36π = 6π
Answer: 6π
Common slips
Slip: Andrew used the whole circle A sector is only a fraction of the circle.
Slip: Andrew used 2πr Area uses πr^2, not circumference.
Slip: Andrew wrote 36π Take 1/6 of 36π to get 6π.
Example 2
Work out the arc length of a sector with angle 90 degrees and radius 4. Give your answer in terms of π.
2 marks
Find fraction of circle: 90 / 360 = 1/4
Calculate full circumference: 2π × 4 = 8π
Multiply fraction by circumference: (1/4) × 8π = 2π
Answer: 2π
Common slips
Slip: Barbara used 1/2 90 degrees is one quarter of 360 degrees.
Slip: Boris calculated area 16π Arc length is a fraction of circumference 2πr.
Slip: Barbara added radii The question asks only for the arc length.
Example 3
Work out the perimeter of a sector with angle 90 degrees and radius 4. Give your answer in terms of π.
3 marks
Calculate arc length: (90/360) × 2π(4) = 2π
Add two straight radii: 4 + 4 = 8
Write total perimeter: 8 + 2π
Answer: 8 + 2π
Common slips
Slip: Chloe stopped at the arc 2π Perimeter includes all bounding edges.
Slip: Clifford added only one radius A sector is enclosed by the arc and two radii.
Slip: Clifford added 8 + 2 = 10 Do not combine numbers and terms in π.
Questions and answers
GCSE Maths sectors and arcs questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Work out the area of a sector with angle 45 degrees and radius 4. Give your answer in terms of π.
Show answer
Answer: 2π
Fraction: 45/360 = 1/8.
Area: (1/8) × 16π = 2π.
M1 — Fraction: 45/360 = 1/8.
A1 — Area: (1/8) × 16π = 2π.
Question 2 Easy2 marks
Work out the area of a sector with angle 90 degrees and radius 2. Give your answer in terms of π.
Show answer
Answer: π
Fraction: 90/360 = 1/4.
Area: (1/4) × 4π = π.
M1 — Fraction: 90/360 = 1/4.
A1 — Area: (1/4) × 4π = π.
Question 3 Medium2 marks
Work out the arc length of a sector with angle 90 degrees and radius 6. Give your answer in terms of π.
Show answer
Answer: 3π
Fraction: 90/360 = 1/4.
Circumference: (1/4) × 12π = 3π.
M1 — Fraction: 90/360 = 1/4.
A1 — Circumference: (1/4) × 12π = 3π.
Question 4 Medium2 marks
Work out the arc length of a sector with angle 60 degrees and radius 3. Give your answer in terms of π.
Show answer
Answer: π
Fraction: 60/360 = 1/6.
Circumference: (1/6) × 6π = π.
M1 — Fraction: 60/360 = 1/6.
A1 — Circumference: (1/6) × 6π = π.
Question 5 Hard2 marks
What fraction of a circle is a 90 degree sector?
Show answer
Answer: 1/4
Angle over 360: 90 / 360.
Simplify: 1/4.
M1 — Angle over 360: 90 / 360.
A1 — Simplify: 1/4.
Question 6 Hard2 marks
What fraction of a circle is a 120 degree sector?
Show answer
Answer: 1/3
Angle over 360: 120 / 360.
Simplify: 1/3.
M1 — Angle over 360: 120 / 360.
A1 — Simplify: 1/3.
Question 7 Exam-style3 marks
Work out the perimeter of a sector with angle 180 degrees and radius 3. Give your answer in terms of π.
Show answer
Answer: 6 + 3π
Arc length is (180/360) × 2π(3) = 3π.
Add two radii: 3 + 3 + 3π = 6 + 3π.
M1 — (180/360) × 2π × 3, or arc length 3π
M1 — Adds two radii to the arc length
A1 — 6 + 3π
Common mistakes
Most lost marks on this topic are one slip.
Using 2πr instead of πr^2 for sector area
Sector area is a fraction of area (πr^2); arc is a fraction of circumference (2πr)
Giving only the arc when perimeter is asked
Sector perimeter includes both straight radii plus the curved arc
Adding numbers to terms in π (e.g. writing 8 + 2π as 10π)
Keep rational terms and π terms separate
Spot the slip
Work out the area of a sector with angle 60 degrees and radius 6. Ada uses the whole circle. What is the correct sector area?
6π
36π
12π
Show the slip
6π 60/360 is one sixth, and one sixth of 36π is 6π.
Exam tips
This topic can appear on a calculator or a non-calculator paper. Show enough working for a method mark.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Write the fraction 360θ first, simplified, then the formula — two clear method lines.
Answers in terms of π are often accepted or even required; only convert when the question asks for a decimal.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.