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Arc Length

HigherHigher tier onlyAQAEdexcelOCR

Master arc length for GCSE Maths with structured, exam-style practice. This Higher resource covers calculating arc length and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Arc length is (angle ÷ 360) of the full circumference.

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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.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

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WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

An arc is part of the circumference of a circle. Its length is a fraction of the whole circumference, and the fraction comes from the angle at the centre.

Since a full circle is \(360^\circ\), an arc subtending \(\theta\) degrees is \(\frac{\theta}{360}\) of the circumference. The formula is therefore \(\text{arc length} = \frac{\theta}{360} \times \pi d\).

The same fraction idea covers every part of a circle. Once you can find the fraction, arc length and sector area work identically — only the formula you multiply by changes.

Revision notes

The formula

Arc length is \(\frac{\theta}{360} \times \pi d\), or equivalently \(\frac{\theta}{360} \times 2\pi r\).

For a \(90^\circ\) arc on a circle of radius \(8\)cm: \(\frac{90}{360} \times 2\pi \times 8 = 4\pi\)cm.

Finding the fraction

Divide the centre angle by \(360\) and simplify.

A \(60^\circ\) angle gives \(\frac{1}{6}\), a \(90^\circ\) angle gives \(\frac{1}{4}\), and \(120^\circ\) gives \(\frac{1}{3}\). Recognising these speeds the work considerably.

Arc length versus sector perimeter

The arc is only the curved part. The perimeter of a sector also includes the two straight radii.

So a sector of radius \(8\)cm with a \(4\pi\)cm arc has perimeter \(4\pi + 16\)cm.

Key points

  • An arc is part of the circumference.
  • Arc length is \(\frac{\theta}{360} \times \pi d\).
  • The fraction comes from the centre angle over 360.
  • A quarter circle has a \(90^\circ\) angle.
  • A sector's perimeter includes two radii.
  • Arc length is in ordinary units, not squared.

Worked examples

Example 1

Find the arc length for a \(90^\circ\) angle on a circle of radius \(6\)cm, in terms of \(\pi\).

Working

\[\frac{90}{360} = \frac{1}{4}\]find the fraction of the circle
\[\frac{1}{4} \times 2\pi \times 6\]multiply by the circumference
\[= 3\pi\text{cm}\]state the arc length

Example 2

Find the arc length for a \(60^\circ\) angle on a circle of diameter \(18\)cm.

Working

\[\frac{60}{360} = \frac{1}{6}\]find the fraction
\[\frac{1}{6} \times 18\pi\]multiply by the circumference
\[= 3\pi\text{cm}\]state the arc length

Example 3

Find the perimeter of a sector with radius \(6\)cm and arc length \(3\pi\)cm.

Working

\[3\pi\]the curved edge
\[+ 6 + 6\]add the two radii
\[= (3\pi + 12)\text{cm}\]state the perimeter

Common mistakes

  • Forgetting the fraction.

    An arc is only part of the circumference, so the angle fraction is essential.

  • Using the radius in the πd formula.

    Either double the radius or use 2πr.

  • Omitting the radii from a sector perimeter.

    The perimeter includes both straight edges as well as the arc.

  • Using squared units.

    Arc length is a length, so cm not cm².

Exam tips

  • Work out the fraction of the circle first and simplify it.
  • Learn the common fractions for 60, 90 and 120 degrees.
  • Add both radii when the question asks for a sector's perimeter.
  • Leave answers in terms of π when exact values are wanted.

Key terms

Arc
Part of the circumference of a circle.
Sector
A slice of a circle bounded by two radii and an arc.
Subtend
To form an angle at the centre.
Centre angle
The angle at the centre of the circle.

Related topics

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