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Area and Circumference of Circles

What you'll learn

  • Name important parts of a circle, like radius, diameter, tangent and sector.
  • Use formulas for circumference and area.
  • Give answers rounded or in terms of π\piπ.
  • Solve simple questions about semi-circles, shaded regions and costs.

1. Circle vocabulary

Before using formulas, make sure you know the circle words. These often appear in short 1-mark questions.

Definition

Key circle words

  • The centre is the point in the middle of a circle.

  • A radius is a straight line from the centre to the edge of the circle.

  • A diameter is a straight line from one side of the circle to the other, passing through the centre.

  • The circumference is the distance all the way around the circle.

  • A tangent is a straight line that touches a circle at exactly one point.

  • An arc is part of the circumference.

  • A sector is a slice of a circle, made from two radii and an arc.

  • A semi-circle is half of a circle.

Key parts of a circle: centre, radius, diameter, circumference, tangent, arc, sector and semi-circle.

Key Idea

Radius and diameter

The diameter is twice the radius: d=2rd = 2rd=2r. The radius is half the diameter: r=d2r = \frac{d}{2}r=2d​.

Example

Naming parts of a circle

  1. If a line goes from the centre to the edge, it is a radius.

  2. If a line goes from edge to edge through the centre, it is a diameter.

  3. If a line just touches the circle once and does not cut through it, it is a tangent.

  4. If a shaded part looks like a pizza slice, it is a sector.

2. Circumference: distance around the edge

Circumference is the perimeter of a circle. You use it when a question says things like “around the edge”, “around the field” or “around the circle”.

Key Idea

Circumference formulas

Use C=πdC = \pi dC=πd if you know the diameter. Use C=2πrC = 2\pi rC=2πr if you know the radius.

Example

Finding circumference from the radius

A circular badge has radius 7.3 cm. Work out its circumference to 2 decimal places.

Circular badge with radius 7.3 cm; the circumference is the distance around the outer edge.

  1. The question gives the radius, so use the radius formula.

    C=2πrC = 2\pi rC=2πr

  2. Substitute r=7.3r = 7.3r=7.3.

    C=2×π×7.3=14.6πC = 2 \times \pi \times 7.3 = 14.6\piC=2×π×7.3=14.6π

  3. Use your calculator.

    C≈45.8672C \approx 45.8672C≈45.8672

  4. Round to 2 decimal places: 45.87 cm.

Example

Cost around a circular field

A circular field has diameter 30 m. Fencing costs £14.50 per metre. Find the total cost.

Circular field with diameter 30 m and fencing needed all the way around its circumference.

  1. “Around the field” means circumference.

  2. The diameter is given, so use C=πdC = \pi dC=πd.

    C=30π≈94.2478C = 30\pi \approx 94.2478C=30π≈94.2478

  3. Multiply the length by the cost per metre.

    94.2478×14.50≈1366.5994.2478 \times 14.50 \approx 1366.5994.2478×14.50≈1366.59

  4. The total cost is £1366.59.

Tip

Use the pi button

Use the π\piπ button on your calculator for decimal answers. Only use 3.14 if the question specifically tells you to.

3. Area: space inside the circle

Area means the amount of space inside a shape. Circle area is measured in square units, such as cm² or m².

Key Idea

Area formula

Use A=πr2A = \pi r^2A=πr2. This formula needs the radius, not the diameter.

Example

Finding area from the diameter

A circle has diameter 11 m. Work out its area correct to 1 decimal place.

Circle with diameter 11 m, showing that the radius needed for the area formula is half the diameter.

  1. Halve the diameter to find the radius.

    r=112=5.5r = \frac{11}{2} = 5.5r=211​=5.5

  2. Use the area formula.

    A=πr2A = \pi r^2A=πr2

  3. Substitute r=5.5r = 5.5r=5.5.

    A=π×5.52=30.25πA = \pi \times 5.5^2 = 30.25\piA=π×5.52=30.25π

  4. Use your calculator.

    A≈95.0332A \approx 95.0332A≈95.0332

  5. Round to 1 decimal place: 95.0 m².

Common Mistake

Using the diameter as the radius

If the diameter is given, halve it before using A=πr2A = \pi r^2A=πr2. Forgetting this makes the area much too large.

4. Answers in terms of pi

Sometimes the question says “give your answer in terms of π\piπ”. This means leave π\piπ in your answer and do not turn it into a decimal.

Example

Leaving answers in terms of pi

  1. A circle has diameter 16 mm. Its circumference is:

Circle with diameter 16 mm for a circumference answer left in terms of pi.

$$
C = \pi d = 16\pi
$$

2. So the circumference is 16π mm.

  1. A circle has radius 9 cm. Its area is:

Circle with radius 9 cm for an area answer left in terms of pi.

$$
A = \pi r^2 = \pi \times 9^2 = 81\pi
$$

4. So the area is 81π cm².

5. Semi-circles and sectors

For part-circles, first think: “What fraction of the full circle do I have?”

  • A semi-circle is one half.
  • A quarter circle is one quarter.
  • Three quarters of a circle is three quarters of the full circle.
Key Idea

Perimeter of part-circles

For curved edges, take the correct fraction of the circumference. For perimeter, remember to add any straight outside edges too.

Example

Perimeter of a semi-circle from its area

A semi-circle has area 40 m². Find its perimeter to 1 decimal place.

Semi-circle of area 40 m², where the perimeter includes both the curved edge and the straight diameter.

  1. A semi-circle is half a circle, so the full circle area would be 80 m².

  2. Use the circle area formula to find the radius.

    πr2=80\pi r^2 = 80πr2=80

  3. Divide by π\piπ and square root.

    r=80π≈5.05r = \sqrt{\frac{80}{\pi}} \approx 5.05r=π80​​≈5.05

  4. The curved edge of a semi-circle is half the circumference.

    π×5.05≈15.86\pi \times 5.05 \approx 15.86π×5.05≈15.86

  5. The diameter is two radii, so it is about 10.10 m.

  6. Perimeter equals curved edge plus diameter: about 26.0 m.

Common Mistake

Forgetting the straight edge

A semi-circle perimeter is not just half the circumference. You must also add the diameter.

6. Shaded and compound areas

A compound shape is made from two or more simple shapes joined together. For shaded regions, you often need to subtract one area from another.

Key Idea

Add or subtract areas

For joined shapes, add the areas. For cut-out or shaded-gap shapes, subtract the smaller area from the larger area.

Example

Area of a shaded ring

A shaded ring is made from a large circle with diameter 18 cm and a smaller circle of radius 7 cm cut out. Find the shaded area to 1 decimal place.

Shaded ring formed by subtracting a smaller circle of radius 7 cm from a larger circle of diameter 18 cm.

  1. Find the large radius by halving the diameter.

    r=182=9r = \frac{18}{2} = 9r=218​=9

  2. Find both circle areas.

    Alarge=π×92=81πAsmall=π×72=49π\begin{aligned} A_{\text{large}} &= \pi \times 9^2 = 81\pi \\ A_{\text{small}} &= \pi \times 7^2 = 49\pi \end{aligned}Alarge​Asmall​​=π×92=81π=π×72=49π​

  3. Subtract the smaller area from the larger area.

    Ashaded=81π−49π=32πA_{\text{shaded}} = 81\pi - 49\pi = 32\piAshaded​=81π−49π=32π

  4. Convert to a decimal and round: 100.5 cm².

Tip

Boxes, tiles and bags

If you need a whole number of boxes, divide the total area by the coverage per box, then round up.

Exam technique

In the exam

  1. Label the radius and diameter on the diagram before choosing a formula.

  2. Decide whether you need a length around the edge or an area inside.

  3. Keep extra calculator digits until the final rounding, and include the correct units.

Self review

Check yourself

  • What formula would you use if the diameter is given and you need the circumference?

  • Why must you halve the diameter before using the area formula?

  • For a semi-circle perimeter, which straight edge must be added to the curved edge?

Recap questions

Test yourself with 5 quick questions on this guide. Answer them all correctly to complete it.

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Area and Circumference of Circles Revision Guide

  1. GCSE
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