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Area of a Sector

HigherHigher tier onlyAQAEdexcelOCR

Area of a Sector is a key geometry topic at GCSE Maths. This Higher worksheet gives you exam-style questions on calculating the area of a sector, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. Sector area is (angle ÷ 360) of the whole circle's area.

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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.

Higher

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Topic overview

A sector is a slice of a circle, bounded by two radii and an arc. Its area is the same fraction of the circle's area as the angle is of a full turn.

The formula is \(\text{sector area} = \frac{\theta}{360} \times \pi r^2\). The structure exactly matches arc length, with the circle's area replacing its circumference.

That parallel is worth noticing. Once you have the fraction, you multiply by \(\pi r^2\) for area or by \(\pi d\) for arc length. Confusing the two is the usual error, and the units give it away: area must be squared.

Revision notes

The formula

Sector area is \(\frac{\theta}{360} \times \pi r^2\).

For a \(90^\circ\) sector of radius \(6\)cm: \(\frac{1}{4} \times \pi \times 36 = 9\pi\)cm².

Comparing with arc length

The fraction is identical; only what you multiply by changes.

Use \(\pi r^2\) for area and \(2\pi r\) for arc length. Checking the units confirms which you have found.

Segments

A segment is the region between a chord and the arc, found by subtracting a triangle from the sector.

Sector area minus the area of the triangle formed by the two radii gives the segment.

Key points

  • A sector is a slice bounded by two radii and an arc.
  • Sector area is \(\frac{\theta}{360} \times \pi r^2\).
  • The fraction matches that used for arc length.
  • Area is in squared units.
  • A quarter sector uses \(\frac{1}{4}\).
  • A segment is a sector minus a triangle.

Worked examples

Example 1

Find the area of a \(90^\circ\) sector of radius \(8\)cm, in terms of \(\pi\).

Working

\[\frac{90}{360} = \frac{1}{4}\]find the fraction of the circle
\[\frac{1}{4} \times \pi \times 8^2\]multiply by the circle's area
\[= 16\pi\text{cm}^2\]state the area

Example 2

Find the area of a \(120^\circ\) sector of radius \(6\)cm, in terms of \(\pi\).

Working

\[\frac{120}{360} = \frac{1}{3}\]find the fraction
\[\frac{1}{3} \times \pi \times 36\]multiply by the area
\[= 12\pi\text{cm}^2\]state the area

Example 3

Find the area of a \(45^\circ\) sector of radius \(4\)cm, in terms of \(\pi\).

Working

\[\frac{45}{360} = \frac{1}{8}\]find the fraction
\[\frac{1}{8} \times \pi \times 16\]multiply by the area
\[= 2\pi\text{cm}^2\]state the area

Common mistakes

  • Using the arc length formula for area.

    Area uses πr², arc length uses 2πr. Check the units to tell them apart.

  • Forgetting the angle fraction.

    A sector is only part of the circle, so the fraction is essential.

  • Using the diameter instead of the radius.

    The area formula needs the radius, so halve the diameter first.

  • Giving area in ordinary units.

    Sector area needs cm² or m².

Exam tips

  • Write the fraction down and simplify it before multiplying.
  • Use πr² for area, never 2πr.
  • Check your units: area must be squared.
  • Leave answers in terms of π when exact values are wanted.

Key terms

Sector
A slice of a circle bounded by two radii and an arc.
Segment
The region between a chord and an arc.
Radius
The distance from the centre to the edge.
Centre angle
The angle at the centre defining the sector.

Related topics

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