Maths
What is the circumference of a circle, and how do you find it?
Quick answer
The circumference is the distance all the way around a circle. C = πd or C = 2πr, so a circle with a diameter of 10 cm has a circumference of about 31.4 cm.
In one sentence
The circumference is the length of the edge of a circle: how far you would walk if you went once all the way around it.
Explained
The circumference is the perimeter of a circle. If you wrapped a piece of string around a plate and then straightened it out, the length of the string would be the plate's circumference. For the names of the other parts (centre, radius, diameter), see parts of a circle.
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Here is the amazing fact behind the formula. Take any circle, big or small, and divide its circumference by its diameter. You always get the same number, a little more than 3. That number is called pi and written π. It is about 3.14 (or 22/7 as a fraction), and its decimals go on forever: see pi. Because circumference ÷ diameter = π, it follows that
C = π × d (circumference = pi × diameter)
The diameter is twice the radius (d = 2r), so you can also write
C = 2 × π × r = 2πr
Both formulas give the same answer. Use πd when you know the diameter and 2πr when you know the radius. Circumference is a length, so the answer is in length units: mm, cm, m or km, never cm².
A good way to check an answer: the circumference is always a bit more than three times the diameter. A wheel 50 cm across is about 3 × 50 = 150 cm around (more exactly, 157 cm). If your answer is far from that, look for a mistake.
You can also go backwards. If you know the circumference, divide by π to get the diameter: d = C ÷ π. Halve that to get the radius, or use r = C ÷ (2π).
Do not mix up the circumference with the area of a circle. The circumference measures the edge (2πr, in cm). The area measures the space inside (πr², in cm²).
Examples
| Given | Diameter | Circumference (π ≈ 3.14) |
|---|---|---|
| radius 1 cm | 2 cm | 3.14 × 2 = 6.28 cm |
| radius 3 cm | 6 cm | 3.14 × 6 = 18.84 cm |
| diameter 10 cm | 10 cm | 3.14 × 10 = 31.4 cm |
| radius 7 cm | 14 cm | 3.14 × 14 = 43.96 cm (44 cm with π ≈ 22/7) |
| diameter 1 m | 1 m | 3.14 m |
| radius 12 m | 24 m | 3.14 × 24 = 75.36 m |
Worked example: a bike wheel has a diameter of 70 cm. In one full turn the bike moves forward one circumference: C = π × 70 ≈ 3.14 × 70 = 219.8 cm, about 2.2 m. After 100 turns it has gone about 220 m.
Common mistakes
- Using the radius in πd. C = πd needs the diameter. If you only have the radius, double it first, or use 2πr. With r = 5 cm, the answer is 31.4 cm, not 15.7 cm.
- Mixing up circumference and area. 2πr is the distance around; πr² is the space inside. A quick check: area has a "squared", circumference does not.
- Writing square units. A circumference is a length: write cm or m, not cm².
- Forgetting the straight edge. The perimeter of a semicircle is half the circumference plus the diameter.
Step by step
Step by step
- Read what you are givenIs it the radius (centre to edge) or the diameter (all the way across)?
- Get the diameterIf you have the radius, double it: d = 2r. (Or keep r and use C = 2πr.)
- Multiply by πUse 3.14, 22/7 or the π button on a calculator, whichever the question asks for.
- Round and add unitsRound as the question says and write a length unit: cm, m, km.
- CheckIs your answer a little more than 3 times the diameter? Then it is sensible.
Exact answers: some questions say "give your answer in terms of π". Then you leave π as a letter. A circle with radius 6 cm has C = 2 × π × 6 = 12π cm.
Practice questions
Find the circumference (use π ≈ 3.14 unless told otherwise)
Try each one, then check your answer.
- Find the circumference of a circle with diameter 4 cm.
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C = 3.14 × 4 = 12.56 cm - Find the circumference of a circle with diameter 20 cm.
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C = 3.14 × 20 = 62.8 cm - Find the circumference of a circle with radius 5 cm.
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C = 2 × 3.14 × 5 = 31.4 cm - Find the circumference of a circle with radius 2.5 m.
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C = 2 × 3.14 × 2.5 = 15.7 m - Find the circumference of a circle with radius 7 cm, using π ≈ 22/7.
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C = 2 × 22/7 × 7 = 44 cm - Find the circumference of a circle with diameter 14 m, using π ≈ 22/7.
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C = 22/7 × 14 = 44 m - A circle has radius 6 cm. Give its circumference in terms of π.
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C = 2 × π × 6 = 12π cm - A circle has diameter 9 cm. Give its circumference in terms of π.
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C = π × 9 = 9π cm - A circle has a circumference of 31.4 cm. Find its diameter.
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d = 31.4 ÷ 3.14 = 10 cm - A circle has a circumference of 18.84 m. Find its radius.
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d = 18.84 ÷ 3.14 = 6 m, so r = 3 m - A circle has a circumference of 100 cm. Find its radius to 1 decimal place (use the π button).
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r = 100 ÷ (2π) = 15.915… ≈ 15.9 cm - Find the perimeter of a semicircle with diameter 10 cm.
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Half the circumference is 31.4 ÷ 2 = 15.7 cm; add the diameter: 15.7 + 10 = 25.7 cm - Find the perimeter of a quarter circle with radius 4 cm.
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Curved part: 2 × 3.14 × 4 ÷ 4 = 6.28 cm; add two radii: 6.28 + 4 + 4 = 14.28 cm - Circle A has radius 3 cm and circle B has radius 6 cm. How many times longer is the circumference of B?
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A: 6π cm, B: 12π cm, so B is 2 times as long - A circle has a circumference of 12π cm. What is its area?
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2πr = 12π, so r = 6 cm; area = π × 6² = 36π ≈ 113.04 cm²
Word problems
Circumference in real life
Try each one, then check your answer.
- A bike wheel has a diameter of 70 cm. How many full turns does it make in 1 km?
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One turn = 3.14 × 70 = 219.8 cm; 1 km = 100,000 cm; 100,000 ÷ 219.8 ≈ 454.96, so about 455 turns - Ribbon is glued around the edge of a round table with diameter 1.2 m. How much ribbon is needed?
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C = 3.14 × 1.2 = 3.768 m (about 3.77 m) - The trunk of a tree measures 157 cm around. What is its diameter?
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d = 157 ÷ 3.14 = 50 cm - A round pond has a radius of 3 m. A fence around it costs $12 per metre. What does the fence cost?
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C = 2 × 3.14 × 3 = 18.84 m; 18.84 × 12 = $226.08 - The Earth's equator is about 40,075 km long. Estimate the Earth's radius.
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r = 40,075 ÷ (2 × 3.14) ≈ 6,381 km (with the π button about 6,378 km)
Questions people ask
What is the formula for the circumference of a circle?
C = πd, where d is the diameter, or C = 2πr, where r is the radius. They are the same formula, because the diameter is twice the radius.
How do you find the circumference with the diameter?
Multiply the diameter by π. A circle with a diameter of 8 cm has C = 3.14 × 8 = 25.12 cm.
How do you find the circumference with the radius?
Multiply the radius by 2 and then by π. A circle with a radius of 7 cm has C = 2 × 3.14 × 7 = 43.96 cm, or 44 cm with π ≈ 22/7.
What units is circumference measured in?
In length units: millimetres, centimetres, metres, kilometres (or inches and feet). Use the same unit as the radius or diameter you were given.
What is the difference between circumference and area?
The circumference is the distance around the circle (2πr). The area is the flat space inside it (πr², in square units). A circle with radius 10 cm has a circumference of 62.8 cm and an area of 314 cm².
How do you measure the circumference of a real object?
Wrap a string or a tape measure tightly around it once, mark where it meets, and measure the string. Or measure straight across through the middle to get the diameter and multiply by 3.14.
How do you find the radius from the circumference?
Divide the circumference by 2π. If C = 31.4 cm, then r = 31.4 ÷ 6.28 = 5 cm.
Quick quiz
Circumference of a circle
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Question 1 of 5
Which formula gives the circumference of a circle with radius r?
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Related topics
Pi (π)
Pi (π) is the number you get when you divide any circle's circumference by its diameter. It is about 3.14, and its digits go on forever without repeating.
Parts of a circle
A circle has a centre, a radius, a diameter and a circumference. The radius runs from the centre to the edge, and the diameter is twice the radius.
Area of a circle
The area of a circle is pi times the radius squared: A = πr². A circle with radius 5 cm has an area of about 78.5 square centimetres.
Perimeter
Perimeter is the total distance all the way around the outside of a flat shape. For a rectangle 5 cm long and 3 cm wide, it is 5 + 3 + 5 + 3 = 16 cm.
Angles in parallel lines
When a line crosses two parallel lines, corresponding (F) and alternate (Z) angles are equal and co-interior (C) angles add to 180°. Vertically opposite angles are always equal.
Area of a trapezium (trapezoid)
Add the two parallel sides, multiply by the height and halve: A = ½(a + b)h. Parallel sides of 4 cm and 6 cm, 3 cm apart, give an area of 15 cm².