Before using any circle formulae, it helps to know the key vocabulary — several GCSE questions are really just testing whether you can identify these parts correctly.
Radius — the distance from the centre to the edge
Diameter — the distance across the circle through the centre (twice the radius)
Circumference — the distance around the outside of the circle
Chord — a straight line connecting two points on the circumference (not through the centre)
Tangent — a straight line that touches the circle at exactly one point
The tangent touches the circle at a single point and is always perpendicular to the radius drawn to that point.
Circumference and area
Circumference = π × diameter = 2πr
Area = πr²
Worked Example 1 — Circumference
Find the circumference of a circle with radius 6cm. Give your answer in terms of π.
1
Circumference = 2πr = 2 × π × 6
2
Circumference = 12π
Answer12π cm (≈ 37.7cm)
Worked Example 2 — Area
Find the area of a circle with radius 9cm. Give your answer to 1 decimal place.
1
Area = πr² = π × 9²
2
Area = π × 81 = 254.47...
Answer254.5cm²
Arc length and sector area
A sector is a "pie slice" of a circle, bounded by two radii and an arc. Both the arc length and the sector area are simply that fraction of the full circumference or area, based on the angle at the centre.
Arc length = (θ ÷ 360) × 2πr
Sector area = (θ ÷ 360) × πr²
Worked Example 3 — Arc length
Find the length of the arc of a sector with radius 9cm and an angle of 120° at the centre.
The sector (shaded) is bounded by two radii and the arc between them — both scale with θ ÷ 360.
Compound circle shapes
Some GCSE questions combine circles (or parts of circles) with other shapes. As with any compound shape, split it into parts you already know how to work with.
Worked Example 4 — Semicircle perimeter
Find the perimeter of a semicircle with radius 5cm.
1
The perimeter has two parts: the curved arc (half the circumference) and the straight diameter.
2
Arc = ½ × 2πr = ½ × 2π × 5 = 5π. Diameter = 2 × 5 = 10
3
Perimeter = 5π + 10 = 25.7cm (1 d.p.)
Answer25.7cm
A semicircle's perimeter is the curved arc plus the straight diameter — not the arc alone.
Practice questions
Work through each question before checking the answers. Where a question describes a sector, compound shape, or annulus, the diagram shown is required to answer it. Use π = 3.14 or the π button on your calculator, and round to 1 decimal place unless told otherwise.
Foundation (Grade 3–5)
Q1Find the circumference of a circle with radius 5cm.Foundation
Q2Find the circumference of a circle with diameter 14cm.Foundation
Q3Find the area of a circle with radius 6cm.Foundation
Q4Find the area of a circle with diameter 10cm.Foundation
Q5A circle has a circumference of 31.4cm. Find its radius.Foundation
Higher (Grade 5–7)
Q6The diagram shows a sector. Find the length of the arc.Higher
Diagram for Question 6 (not to scale).
Q7Find the area of a sector with radius 10cm and an angle of 60° at the centre.Higher
Q8Find the length of the arc of a sector with radius 12cm and an angle of 45° at the centre.Higher
Q9The diagram shows a semicircle. Find its perimeter.Higher
Diagram for Question 9 (not to scale).
Q10A sector with radius 6cm has an area of 20cm². Find the angle at the centre.Higher
Higher — Hard (Grade 8–9)
Q11The diagram shows a window made from a rectangle with a semicircle on top. Find the total perimeter of the window.Grade 8–9
Diagram for Question 11 (not to scale).
Q12The diagram shows an annulus (a ring shape) formed by two circles sharing the same centre. Find the shaded area.Grade 8–9
Diagram for Question 12 (not to scale). The outer radius (9cm) and inner radius (5cm) are shown separately for clarity — the shaded ring between the two circles is the annulus.
Q13A sector with an angle of 50° has an area of 15cm². Find the radius (1 d.p.).Grade 8–9
Q14An arc has length 5π cm and radius 10cm. Find the angle x at the centre.Grade 8–9
Q15The diagram shows a square of side 8cm with a quarter-circle of radius 8cm removed from one corner. Find the remaining shaded area.Grade 8–9
Diagram for Question 15 (not to scale). The light grey corner (radius 8cm) is removed.
Answers
Foundation (Q1–Q5)
Q131.4cm(2π × 5)
Q244.0cm(π × 14)
Q3113.1cm²(π × 6²)
Q478.5cm²(radius 5, π × 5²)
Q55.0cm(31.4 ÷ (2 × 3.14))
Higher (Q6–Q10)
Q612.6cm(¼ × 2π × 8)
Q752.4cm²(⅙ × π × 10²)
Q89.4cm(⅛ × 2π × 12)
Q936.0cm(π × 7 + 14)
Q1063.7°(20 × 360 ÷ (36π))
Higher — Hard (Q11–Q15)
Q1155.7cm(2 × 15 + 10 + π × 5)
Q12175.9cm²(π × (9² − 5²))
Q135.9cm(r² = 15 × 360 ÷ (50π))
Q14x = 90°(5π = (x÷360) × 20π)
Q1513.7cm²(8² − ¼π × 8²)
Common mistakes
Common Mistake 1
Using the diameter instead of the radius (or vice versa)
Area always uses the radius (Area = πr²) — if you're given a diameter, halve it first. Circumference can use either (C = πd or C = 2πr), so check which one you're substituting into.
Common Mistake 2
Forgetting the diameter in a semicircle's perimeter
A semicircle's perimeter is the curved arc plus the straight diameter — forgetting the straight edge is one of the most common errors with this shape.
Common Mistake 3
Using the wrong fraction for arc length or sector area
Always divide the given angle by 360° first — using the angle itself (rather than θ ÷ 360) in the formula is a very common slip.
Common Mistake 4
Subtracting the wrong way round in an annulus
Annulus area = π × (outer radius² − inner radius²) — subtract the areas, not the radii, and make sure the larger circle's area comes first.
Exam tips
💡 Exam Tip 1
Leave answers in terms of π when asked
If a question says "give your answer in terms of π", write it as e.g. 12π — don't convert to a decimal, as this loses marks.
💡 Exam Tip 2
Sketch compound shapes and label each part
For semicircles, annuli and other compound shapes, sketch the shape and mark which parts are curved and which are straight before calculating.
💡 Exam Tip 3
Write the formula before substituting
Writing "Arc length = (θ ÷ 360) × 2πr" before putting numbers in secures method marks even if the final calculation goes wrong.
💡 Exam Tip 4
Check your calculator's π button
Using the calculator's π button gives a more accurate answer than using 3.14 — only use 3.14 if the question specifically asks you to.
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