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Circumference

FoundationHigherAQAEdexcelOCR

Practise circumference with this free Foundation and Higher GCSE Maths worksheet from Virtus Academy. You'll work through calculating the circumference of a circle, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. Circumference = πd (or 2πr).

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

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Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

The circumference is the distance around a circle. It is found using \(C = \pi d\), where \(d\) is the diameter, or equivalently \(C = 2\pi r\) using the radius.

The two formulae are the same because the diameter is twice the radius. Choosing the one matching the measurement you are given saves a step and avoids errors.

The most frequent mistake is using the radius in the diameter formula. If the question gives a radius of \(5\)cm, the diameter is \(10\)cm, and using \(5\) in \(\pi d\) halves the answer.

Revision notes

The formulae

\(C = \pi d\) with the diameter, or \(C = 2\pi r\) with the radius.

For a circle of radius \(7\)cm: \(C = 2 \times \pi \times 7 = 44.0\)cm to one decimal place.

Exact answers in terms of pi

Leaving \(\pi\) in the answer keeps it exact.

So a circle of diameter \(10\)cm has circumference \(10\pi\)cm exactly, which is more precise than \(31.4\)cm.

Arcs and semicircles

A semicircle's curved edge is half the circumference, but its perimeter also includes the straight diameter.

So a semicircle of diameter \(8\)cm has perimeter \(\frac{8\pi}{2} + 8 = 4\pi + 8\)cm. Forgetting the straight edge is a common error.

Key points

  • Circumference is the distance around a circle.
  • \(C = \pi d\) or \(C = 2\pi r\).
  • The diameter is twice the radius.
  • Leave \(\pi\) in the answer for an exact value.
  • A semicircle's perimeter includes the diameter.
  • Units are ordinary lengths, not squared.

Worked examples

Example 1

Find the circumference of a circle with diameter \(14\)cm, in terms of \(\pi\).

Working

\[C = \pi d\]use the diameter formula
\[= 14\pi\text{cm}\]state the exact circumference

Example 2

Find the circumference of a circle with radius \(5\)cm to 1 decimal place.

Working

\[C = 2\pi r = 2 \times \pi \times 5\]use the radius formula
\[= 10\pi\]simplify
\[= 31.4\text{cm}\]evaluate to one decimal place

Example 3

Find the perimeter of a semicircle with diameter \(12\)cm, in terms of \(\pi\).

Working

\[\frac{12\pi}{2} = 6\pi\]half the circumference gives the curved edge
\[6\pi + 12\]add the straight diameter
\[= (6\pi + 12)\text{cm}\]state the perimeter

Common mistakes

  • Using the radius in \(C = \pi d\).

    Double the radius first, or use C = 2πr instead.

  • Forgetting the diameter on a semicircle perimeter.

    The perimeter includes the straight edge as well as the curve.

  • Rounding too early.

    Keep π in the working and round only at the end.

  • Using squared units.

    Circumference is a length, so cm not cm².

Exam tips

  • Check whether you are given the radius or the diameter.
  • Leave answers in terms of π when the question asks for exact values.
  • Add the straight edge for semicircle perimeters.
  • Use ordinary length units.

Key terms

Circumference
The distance around a circle.
Diameter
A line through the centre, twice the radius.
Radius
The distance from the centre to the edge.
Semicircle
Half a circle.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.