GCSE Maths revision · Geometry and measures

Angles in polygons

Foundation and Higher

The idea

The exterior angles of any polygon add to $360^\circ$ . The interior angles of an $n$ -sided polygon add to $(n - 2) \times 180^\circ$ . At each vertex, interior $+$ exterior $= 180^\circ$ .

For a regular polygon, exterior angle $= 360^\circ \div n$ .
Interior angle $= 180^\circ -$ exterior angle.
Number of sides $= 360^\circ \div$ exterior angle.

Watch out $360 \div n$ gives the exterior angle, not the interior one.

See it worked

Worked exampleNon-calculator3 marks

A decagon has 10 sides. This decagon is regular.

(a) Work out the size of each exterior angle.

(b) Work out the size of each interior angle.

(a) The exterior angles of any polygon add up to $360^\circ$ .
$360 \div 10 = 36$
(b) An interior and an exterior angle sit on a straight line, so they add up to $180^\circ$ .
$180 - 36 = 144$

Answer (a) $36^\circ$ ; (b) $144^\circ$

Your turn

Question 1Non-calculator3 marks

Triangle ABC has AB = AC. Angle ABC is (3x + 5) $^\circ$ and angle BAC is $4x^\circ.$ Work out angle BAC.

Question 2Non-calculator4 marks

A regular polygon has 8 sides.

(a) Find one exterior angle.

(b) Find the sum of its interior angles.

Question 3Non-calculator5 marks

Each interior angle of a regular polygon is $156^\circ$ .

(a) Work out the number of sides of the polygon.

(b) Work out the sum of the interior angles of the polygon.

In past papers

144 official questions in 122 papers since 2017. The latest:

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