G3GeometryFoundation & Higher

Circles

Circles have their own special formulas built around π (pi ≈ 3.14). Once you know the circumference and area formulas, arcs and sectors are just fractions of the whole circle.

40 min AQA GCSE Maths

What you'll learn

Find the circumference of a circle
Find the area of a circle
Work with radius and diameter
Find arc length and sector area

Circumference and area

The $circumference$ (distance around) is $π d$ or $2 π r$ . The $area$ is $π r^{2}$ . The radius is half the diameter.

C = π d = 2 π r

Circumference.

A = π r^{2}

Area.

A = π r^{2} = π \times 5^{2}

= 25 π \approx 78.5

cm².

Answer78.5 cm²

Arcs and sectors

A sector is a “pizza slice”. Its arc length and area are fractions of the whole circle: take the angle over $360°$ and multiply by the circumference or area.

arc = \frac{θ}{360} \times π d, sector area = \frac{θ}{360} \times π r^{2}

Fraction of the whole circle.

Tip — Use r for area (r²) and either d or 2r for circumference — don’t mix them up.

Remember these

C = π d

Circumference.

A = π r^{2}

Area.

sector = \frac{θ}{360} π r^{2}

Sector area.

Watch out for these

Using the diameter in the area formula:

A = π d^{2}

Area uses the radius:

A = π r^{2}

Forgetting the radius is half the diameter.

r = d ÷ 2.

Key takeaways

Circumference: C = πd = 2πr.
Area: A = πr².
Arc/sector = (angle ÷ 360) × whole circle.

Test yourself

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