KS3 Maths Angles in Polygons With Examples
Polygon angle questions become manageable when students distinguish interior angles, exterior angles and the number of sides. A triangle has an interior-angle sum of 180°, while a quadrilateral has 360°. Larger polygons follow a consistent pattern: drawing diagonals from one vertex divides a polygon into triangles, giving a clear explanation of the formula.
Why the interior-angle sum increases
From one vertex, a convex polygon with n sides can be divided into n − 2 triangles without overlap. Each triangle contributes 180° to the total, so the sum of interior angles is (n − 2) × 180°. For a pentagon, n = 5 and the sum is 3 × 180° = 540°. For a hexagon, the sum is 4 × 180° = 720°. This reasoning is more memorable than an isolated formula.
Find each interior angle in a regular polygon
A regular polygon has equal sides and equal interior angles. Divide the interior-angle sum by the number of sides. A regular pentagon has each interior angle equal to 540° ÷ 5 = 108°. A regular hexagon has each interior angle equal to 720° ÷ 6 = 120°. Do not divide by the number of sides when the polygon is irregular, because its interior angles need not be equal.
Related: Year 8 Maths Sequences and Nth Terms
Understand exterior angles
An exterior angle is formed by extending one side of a polygon and measuring the angle outside at a vertex. The adjacent interior and exterior angles lie on a straight line and sum to 180°. Moving around a convex polygon in one direction, the exterior turning angles sum to 360°. Thus each exterior angle of a regular octagon is 360° ÷ 8 = 45°, and each interior angle is 180° − 45° = 135°.
Find the number of sides from an exterior angle
If each exterior angle of a regular polygon is 40°, the number of sides is 360° ÷ 40° = 9. Therefore it is a regular nonagon. Check whether the problem asks for the sides or the total sum of interior angles. A common mistake is to use (n − 2) × 180° directly when the exterior angle is given; the 360° relationship is usually quicker.
Solve missing angles in irregular polygons
Suppose four interior angles of a pentagon are 100°, 110°, 95° and 120°. The total is 540°, so the missing angle is 540° − (100° + 110° + 95° + 120°) = 115°. Label the known values on a sketch and make sure you are adding interior angles rather than accidentally including an exterior angle. An irregular polygon can still use the total-sum formula even though its individual angles differ.
Check the shape and the reason
Read whether the polygon is regular. Mark interior or exterior angles clearly and state which theorem you are using. A regular polygon with many sides has small exterior angles and interior angles approaching 180°, which gives a sensible estimate. This intuitive check helps spot answers that are too large or too small. Connecting polygon questions to triangles and straight-line angles strengthens general geometry reasoning.
Frequently asked questions
Does the interior-angle sum formula work for any convex polygon?
Yes, a convex n-sided polygon has interior-angle sum (n − 2) × 180°.
Do exterior angles always add to 360°?
One consistent set of exterior turning angles around a simple convex polygon sums to 360°.
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