Use C=πd and A=πr2 for circles and semicircles, giving answers rounded or in terms of π. It is on Foundation and Higher, aimed at grade 3. Free questions with answers are on this page.
A circle is controlled by its radius r (centre to rim). Diameter d=2r. Circumference (the perimeter) is πd or 2πr. Area is πr2. The r2 is the trap: double the radius and the area quadruples, not doubles.
Leave answers in terms of π when asked (16π cm²) — that is exact. Otherwise use the π button and round at the end to 3 s.f. or 1 d.p. as told. A 'ring' or path around a circle is the large area minus the small area; do not subtract radii and then square.
Radius is half the diameter. C=2πr (or πd) and A=πr2. Leave answers in terms of π on non-calc; 3 s.f. with π=3.142 (or the π button) on calc. Working backwards from area: divide by π, then square-root.
Objectives
The diameter is twice the radius.
C = πd = 2πr.
A = πr².
Key points
Know the difference between radius and diameter and swap between them.
Work out circumference and area of a full circle.
Find the area and perimeter of a semicircle.
Leave answers in terms of π when asked, e.g. 25π cm².
Worked examples
Example 1
The circumference of a circle of radius 2 is a multiple of π. What is that multiple?
2 marks
C = 2πr, so the multiple of π is 2 × 2.
2 × 2 = 4.
The circumference is 4π.
Answer: 4π
Common slips
Slip: C = πr, so the multiple of π is 2. The circumference is 2πr. The 2 has been left out.
Slip: The multiple is 1 × 2 = 2. Without the 2, this is only the radius.
Slip: The circumference is 2π. That is half of the circumference.
Example 2
The area of a circle of radius 3 is a multiple of π. What is that multiple?
2 marks
A = πr², so the multiple of π is 3 × 3.
3 × 3 = 9.
The area is 9π.
Answer: 9π
Common slips
Slip: A = 2πr, so the multiple of π is 2 × 3. The area is πr². The formula 2πr is the circumference.
Slip: 2 × 3 = 6. The radius has to be multiplied by itself, not doubled.
Slip: The area is 6π. That is the circumference multiple, not the area.
Example 3
A circle has diameter 6. Its area is a multiple of π. What is that multiple?
3 marks
The radius is half the diameter: 6 ÷ 2 = 3.
A = πr², so the multiple of π is 3 × 3 = 9.
The area is 9π.
Answer: 9π
Common slips
Slip: A = πr², so the multiple of π is 6 × 6. The diameter has been used where the radius belongs. Halve it first.
Slip: 6 × 6 = 36. This squares the diameter, which is twice the radius.
Slip: The area is 36π. The area is four times too big.
Questions and answers
GCSE Maths area and circumference of circles questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
The circumference of a circle of radius 3 is a multiple of π. What is that multiple?
Show answer
Answer: 6π
C = 2πr, so the multiple of π is 2 × 3.
2 × 3 = 6.
M1 — C = 2πr, so the multiple of π is 2 × 3.
A1 — Answer 6π
Question 2 Easy2 marks
The area of a circle of radius 4 is a multiple of π. What is that multiple?
Show answer
Answer: 16π
A = πr², so the multiple of π is 4 × 4.
4 × 4 = 16.
M1 — A = πr², so the multiple of π is 4 × 4.
A1 — Answer 16π
Question 3 Medium2 marks
The area of a circle of radius 5 is a multiple of π. What is that multiple?
Show answer
Answer: 25π
A = πr², so the multiple of π is 5 × 5.
5 × 5 = 25.
M1 — A = πr², so the multiple of π is 5 × 5.
A1 — Answer 25π
Question 4 Medium2 marks
A circle has diameter 10. Its area is a multiple of π. What is that multiple?
Show answer
Answer: 25π
The radius is half the diameter: 10 ÷ 2 = 5.
A = πr², so the multiple of π is 5 × 5 = 25.
M1 — The radius is half the diameter: 10 ÷ 2 = 5.
A1 — Answer 25π
Question 5 Hard2 marks
A circle has diameter 12. Its area is a multiple of π. What is that multiple?
Show answer
Answer: 36π
The radius is half the diameter: 12 ÷ 2 = 6.
A = πr², so the multiple of π is 6 × 6 = 36.
M1 — The radius is half the diameter: 12 ÷ 2 = 6.
A1 — Answer 36π
Question 6 Hard2 marks
The circumference of a circle of radius 6 is a multiple of π. What is that multiple?
Show answer
Answer: 12π
C = 2πr, so the multiple of π is 2 × 6.
2 × 6 = 12.
M1 — C = 2πr, so the multiple of π is 2 × 6.
A1 — Answer 12π
Question 7 Exam-style2 marks
The circumference of a circle of radius 7 is a multiple of π. What is that multiple?
Show answer
Answer: 14π
C = 2πr, so the multiple of π is 2 × 7.
2 × 7 = 14.
The circumference is 14π.
M1 — C = 2πr, so the multiple of π is 2 × 7.
A1 — Answer 14π
Common mistakes
Most lost marks on this topic are one slip.
Using the diameter in r².
Halve the diameter first.
Squaring π as well as the radius.
Only the radius is squared.
Giving an area in cm.
Area is in cm².
Spot the slip
The radius is
half the diameter
twice the diameter
the distance around the circle
Show the slip
half the diameter The diameter passes through the centre and is two radii.
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 formula you are using before substituting — that line earns a method mark.
'In terms of π' means no decimals: leave 25π, and simplify 21×100π to 50π.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.