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Angles in Polygons

FoundationHigherAQAEdexcelOCR

Angles in Polygons is a key geometry topic at GCSE Maths. This Foundation and Higher worksheet gives you exam-style questions on interior and exterior angles of polygons, 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. Interior angles of an n-sided polygon sum to (n−2) × 180°.

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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

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Topic overview

A polygon's angles follow two rules. The exterior angles of any polygon always add to \(360^\circ\), however many sides it has, and each interior angle sits on a straight line with its exterior angle.

That second fact means an interior and exterior angle at the same vertex add to \(180^\circ\). Together the two rules let you find any angle in a regular polygon quickly.

For a regular polygon with \(n\) sides, each exterior angle is \(\frac{360}{n}\). Finding the exterior angle first and then subtracting from \(180\) is almost always faster than using the interior angle sum formula directly.

Revision notes

Exterior angles

The exterior angles of any polygon add to \(360^\circ\), regardless of the number of sides.

For a regular polygon, each one is \(\frac{360}{n}\). So a regular hexagon has exterior angles of \(360 \div 6 = 60^\circ\).

Interior angles

An interior angle and its exterior angle lie on a straight line, so they add to \(180^\circ\).

For the regular hexagon: \(180 - 60 = 120^\circ\) for each interior angle. This route is quicker than the sum formula.

The interior angle sum

The interior angles of an \(n\)-sided polygon add to \((n - 2) \times 180^\circ\), because the shape splits into \(n - 2\) triangles.

For a pentagon that gives \(3 \times 180 = 540^\circ\). Use this when the polygon is irregular and you need the total.

Key points

  • Exterior angles of any polygon add to \(360^\circ\).
  • Each exterior angle of a regular polygon is \(\frac{360}{n}\).
  • Interior and exterior angles add to \(180^\circ\).
  • Interior angle sum is \((n-2) \times 180^\circ\).
  • The shape splits into \(n - 2\) triangles.
  • Find the exterior angle first for regular polygons.

Worked examples

Example 1

Find each exterior angle of a regular octagon.

Working

\[360 \div 8\]exterior angles add to 360 and there are 8 of them
\[= 45^\circ\]state each exterior angle

Example 2

Find each interior angle of a regular octagon.

Working

\[360 \div 8 = 45^\circ\]find the exterior angle first
\[180 - 45\]interior and exterior add to 180
\[= 135^\circ\]state each interior angle

Example 3

A regular polygon has exterior angles of \(24^\circ\). How many sides does it have?

Working

\[360 \div 24\]divide 360 by the exterior angle
\[= 15 \text{ sides}\]state the number of sides

Common mistakes

  • Dividing 180 by n for the exterior angle.

    Exterior angles come from 360 ÷ n, not 180 ÷ n.

  • Using the interior sum when the exterior route is quicker.

    For regular polygons, find the exterior angle first and subtract from 180.

  • Forgetting the −2 in the sum formula.

    The interior sum is (n − 2) × 180, because the shape splits into n − 2 triangles.

  • Applying regular-polygon rules to an irregular shape.

    Only regular polygons have all angles equal.

Exam tips

  • Find the exterior angle first when the polygon is regular.
  • Use 360 ÷ exterior angle to find the number of sides.
  • Remember the interior sum formula for irregular polygons.
  • Check the interior angle is sensible: it must be under 180.

Key terms

Polygon
A shape with straight sides.
Regular polygon
A polygon with all sides and angles equal.
Interior angle
An angle inside the polygon.
Exterior angle
The angle between a side and the extension of the next side.

Related topics

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