Year 8 · Mathematics · Measurement
Circles: circumference and area
Understand pi as the ratio of circumference to diameter, and calculate the circumference and area of circles and simple parts of circles.
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Learning goals
Learners will
- Identify the radius, diameter and circumference of a circle.
- Understand pi as the ratio of a circle's circumference to its diameter.
- Calculate the circumference and area of circles and semicircles.
Success looks like
- I can explain the link between the radius and the diameter.
- I can calculate a circumference using C = πd or C = 2πr.
- I can calculate an area using A = πr^2, including for a semicircle.
The big idea
Every circle has a centre. The radius (r) is the distance from the centre to the edge, and the diameter (d) is the distance straight across through the centre. The diameter is always twice the radius: d = 2r.
If you measure the circumference and the diameter of any circle and divide the first by the second, you always get the same number, a little more than 3. This number is called pi and written π. Its decimal digits go on forever without repeating, so we usually use the π button on a calculator or the approximation 3.14.
Because circumference ÷ diameter = π, we can write the circumference formula as C = πd. Since d = 2r, this is the same as C = 2πr.
The area of a circle is A = πr^2. One way to see why: cut a circle into many thin wedges and arrange them alternately point-up and point-down. They form a shape close to a rectangle whose length is half the circumference (πr) and whose height is the radius (r), so the area is πr × r = πr^2.
Always check whether you have been given the radius or the diameter. The area formula needs the radius, so halve a diameter first.
Sometimes an exact answer is wanted. Then leave π in the answer, for example a circle of radius 3 cm has area 9π cm^2. Otherwise, round as instructed and include units: cm for lengths and cm^2 for areas.
Worked examples
Circumference from a diameter
Find the circumference of a circle with diameter 10 cm. Use π ≈ 3.14.
- Use C = πd.
- C = 3.14 × 10 = 31.4.
Answer: 31.4 cm
Area from a radius
Find the area of a circle with radius 5 cm. Use π ≈ 3.14.
- Use A = πr^2.
- r^2 = 5 × 5 = 25.
- A = 3.14 × 25 = 78.5.
Answer: 78.5 cm^2
Area of a semicircle
A garden bed is a semicircle with a diameter of 12 m. Find its area. Use π ≈ 3.14.
- Halve the diameter to get the radius: r = 6 m.
- Area of the full circle: 3.14 × 6^2 = 3.14 × 36 = 113.04 m^2.
- A semicircle is half of this: 113.04 ÷ 2 = 56.52.
Answer: 56.52 m^2
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
A circle has a radius of 4 cm. What is its diameter?
8 cm — The diameter is twice the radius: 2 × 4 = 8 cm.
Which formula gives the area of a circle?
A = πr^2 — The area of a circle is A = πr^2. The formulas πd and 2πr give the circumference.
Find the circumference of a circle with diameter 20 cm, using π ≈ 3.14.
62.8 cm — C = πd = 3.14 × 20 = 62.8 cm.
Find the area of a circle with radius 10 m, using π ≈ 3.14.
314 m^2 — A = πr^2 = 3.14 × 100 = 314 m^2.
If the radius of a circle is doubled, what happens to its area?
It becomes four times as large — Area depends on r^2. Doubling r gives (2r)^2 = 4r^2, so the area is four times as large.
Find the circumference of a circle with radius 3 cm, using π ≈ 3.14. Give the number in cm.
18.84 cm — C = 2πr = 2 × 3.14 × 3 = 18.84 cm.
Find the area of a circle with diameter 6 cm, using π ≈ 3.14. Give the number in cm^2.
28.26 cm^2 — The radius is 3 cm, so A = 3.14 × 3^2 = 3.14 × 9 = 28.26 cm^2.
Give the exact area of a circle with radius 7 cm, in terms of π (for example, 4π).
49π cm^2 — A = πr^2 = π × 7^2 = 49π cm^2.
Watch out for
- Using the diameter in A = πr^2. Always check which measurement is given and halve the diameter first.
- Confusing circumference and area formulas. Remind learners that area is measured in square units, which matches the squared r in πr^2.
- Thinking π is exactly 3.14 or exactly 22/7. These are approximations; π is an irrational number.
Tips for parents
- Measure real circles together with a piece of string. Dividing circumference by diameter for several objects is a memorable way to discover π.
- Ask your learner to estimate first ("a bit more than 3 times the diameter") so they can spot answers that are clearly wrong.
- Check that every answer includes the correct units.
Go further
- Measure the circumference and diameter of five round objects, record them in a table, and calculate circumference ÷ diameter for each. How close do you get to π?
- Cut a paper circle into 16 wedges and rearrange them into an approximate rectangle to see why A = πr^2.
- Work out how far a bicycle travels in one turn of a wheel, then how many turns it takes to ride 1 km.