Free GCSE Maths lesson: Geometry and Measures
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Lesson 41 · GCSE / Key Stage 4 · Maths · Geometry and Measures

Polygons and Interior/Exterior Angles

Find angle sums in polygons, work with regular polygons and use exterior angles as a fast route through exam questions.

Qualification: GCSE Key Stage 4 Subject: Maths Strand: Geometry and Measures

Lesson overview

Polygon angle facts turn shape diagrams into calculations.

GCSE questions may ask for a missing angle, a number of sides, or a reason why a regular polygon has a particular angle. The core facts are reliable once you know when to use each one, and the method marks usually come from naming the rule before calculating.

QualificationGCSE Mathematics
Key stageKey Stage 4
StrandGeometry and Measures
Tier Both
Calculator status Calculator and non-calculator
Exam-board status Shared geometry skill

What you will learn

  • How to name common polygons.
  • How to find an interior angle sum using triangles.
  • How to find one interior angle in a regular polygon.
  • Why exterior angles add to 360°.
  • How to solve polygon questions that work backwards.
  • How to use algebraic angle expressions and tiling clues carefully.

Key facts before you start

Interior angle An interior angle is inside the polygon.
Exterior angle Exterior angles of a polygon total 360 degrees.
Interior sum Interior angle sum is (n - 2) x 180 degrees.
Regular polygon A regular polygon has equal sides and equal angles.

Why this matters

Polygon angles appear in shape problems, tiling, bearings, construction-style reasoning and proof. The same two totals keep coming back: interior angles are built from triangles, and exterior angles make one full turn.

Prior knowledge

You should already be comfortable with:

  • triangle angles adding to 180°,
  • angles on a straight line adding to 180°,
  • angles around a point adding to 360°,
  • using simple formulae and division.

Polygons and interior/exterior angles infographic

Before viewing

Before viewing, identify interior and exterior angles on the polygon.

Infographic explaining GCSE Maths polygon angle facts, including interior angle sums, regular polygon angles, exterior angles, reverse methods and common mistakes.
Use this visual to choose the correct polygon angle rule, substitute the number of sides, calculate accurately and check whether the polygon is regular.Download visual

After viewing

After viewing, calculate either an exterior angle or an interior angle sum.

Clear explanation

Interior angle sums

A polygon with n sides can be split into n − 2 triangles from one vertex. Each triangle has 180°, so:

Interior angle sum = (n − 2) × 180°
Pentagon split into three triangles A five-sided polygon is divided from one vertex into three triangles, showing why the interior angle sum is three lots of 180 degrees. 180° 180° 180° 5 sides: (5 − 2) × 180° = 540°
Checked diagram: a pentagon splits into 3 triangles, so its interior angle sum is 540°.

Regular polygons

A regular polygon has all sides equal and all angles equal. To find one interior angle, find the total interior angle sum and divide by the number of sides.

Hexagon interior sum = (6 − 2) × 180° = 720° One interior angle in a regular hexagon = 720° ÷ 6 = 120°

Exterior angles

One exterior angle at each vertex of any convex polygon adds to 360° in total. In a regular polygon:

One exterior angle = 360° ÷ number of sides

Interior and exterior angles on a straight line add to 180°, so one regular interior angle is 180° minus one exterior angle.

If a diagram is not marked as regular, do not divide the interior angle sum by the number of sides. Look for equal-angle marks, equal-side marks or wording that says the polygon is regular.

Some exam questions use algebraic angle labels such as 3x + 10. Add the angles to the correct polygon total first, then solve the equation. In tiling questions, angles meeting at a point must add to 360°.

Worked examples

Example 1: Interior angle sum

Find the interior angle sum of an octagon.

(8 − 2) × 180° = 6 × 180° = 1080°
Reveal answer

Answer: 1080°.

Example 2: One angle in a regular polygon

Find one interior angle of a regular pentagon.

(5 − 2) × 180° = 540° 540° ÷ 5 = 108°
Reveal answer

Answer: 108°.

Example 3: Working backwards from an exterior angle

A regular polygon has exterior angle 45°. How many sides does it have?

360° ÷ 45° = 8
Reveal answer

Answer: 8 sides.

Quick checks

Choose an answer, then check your thinking.

1. The interior angle sum of a quadrilateral is:

2. Exterior angles of a convex polygon add to:

3. A regular polygon with 10 sides has one exterior angle of:

Practice questions

Question 1

Find the interior angle sum of a hexagon.

Reveal answer and marking guidance

Answer: 720°.

Marking: Use (6 − 2) × 180° = 4 × 180°.

Question 2

Find one interior angle of a regular octagon.

Reveal answer and marking guidance

Answer: 135°.

Marking: The interior sum is 1080°, then 1080° ÷ 8 = 135°.

Question 3

Find one exterior angle of a regular 12-sided polygon.

Reveal answer and marking guidance

Answer: 30°.

Marking: Use 360° ÷ 12.

Question 4

A regular polygon has exterior angle 24°. How many sides does it have?

Reveal answer and marking guidance

Answer: 15 sides.

Marking: Use 360° ÷ 24° = 15.

Question 5

Five angles in a hexagon are 110°, 120°, 130°, 140° and 100°. Find the sixth angle.

Reveal answer and marking guidance

Answer: 120°.

Marking: The hexagon total is 720°. The five known angles add to 600°, so the missing angle is 120°.

Question 6

One interior angle of a regular polygon is 150°. How many sides does it have?

Reveal answer and marking guidance

Answer: 12 sides.

Marking: The exterior angle is 180° − 150° = 30°, then 360° ÷ 30° = 12.

Question 7

The angles in a quadrilateral are x°, 2x°, 3x° and 4x°. Find the largest angle.

Reveal answer and marking guidance

Answer: 144°.

Marking: Quadrilateral angles add to 360°, so x + 2x + 3x + 4x = 360. Then 10x = 360, x = 36 and the largest angle is 4x = 144°.

Question 8

Three regular hexagons meet at a point in a tiling. Show that there is no gap.

Reveal answer and marking guidance

Answer: No gap, because 3 × 120° = 360°.

Marking: Find one interior angle of a regular hexagon as 120°, then show that the three angles around the point make a full turn.

Question 9

Two regular octagons and one regular square meet at a point. Is there a gap or overlap?

Reveal answer and marking guidance

Answer: No gap or overlap, because 135° + 135° + 90° = 360°.

Marking: Find one interior angle of a regular octagon as 135° and one square angle as 90°, then compare the total with 360° around a point.

Question 10

Two regular pentagons and one regular polygon meet at a point with no gap or overlap. How many sides does the third polygon have?

Reveal answer and marking guidance

Answer: 10 sides.

Marking: One regular pentagon angle is 108°, so two use 216°. The third interior angle is 360° − 216° = 144°. Its exterior angle is 180° − 144° = 36°, so the number of sides is 360° ÷ 36° = 10.

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For polygon questions, marks usually come from choosing the right total, substituting the number of sides carefully, dividing only when the polygon is regular, forming any algebraic equation clearly, and explaining whether you used interior angles, exterior angles, a straight-line relationship or a full turn around a point. In tiling questions, add the interior angles that meet at the point and compare them with 360°.

Common mistakes

  • Dividing too early: only divide the interior angle sum by n if the polygon is regular.
  • Using n × 180°: the correct interior sum is (n − 2) × 180°.
  • Forgetting exterior angles total 360°: this total does not depend on the number of sides.
  • Mixing interior and exterior angles: one interior angle and its adjacent exterior angle add to 180°.

Extension challenge

A regular polygon has an interior angle that is 5 times its exterior angle. How many sides does it have?

Reveal answer

Answer: 12 sides.

If the exterior angle is x, the interior angle is 5x. They add to 180°, so 6x = 180° and x = 30°. Then 360° ÷ 30° = 12.

Exam-board guidance

Polygon angle facts are assessed across GCSE Maths boards. The common exam habit is to identify whether the polygon is regular, then state the formula or exterior-angle fact before calculating.

AQA GCSE Maths

Focus on the shared method, notation and checking habit; wording and context can vary by route.

OCR GCSE Maths

Show clear method steps and use precise notation; question wording may vary by route.

Pearson Edexcel GCSE Maths

Keep working visible and check units or notation; contexts may vary by route.

Eduqas GCSE Maths

Use the same core skill in practical contexts, with clear reasoning and final checks.

WJEC Wales

Connect the method to the context and state what the result means.

CCEA GCSE Maths

Make each method step visible and keep notation, units and final checks clear.

Next lesson

Next, continue Geometry and Measures with Perimeter and Area.

Previous lesson: Angles and Parallel Lines Next lesson: Perimeter and Area Back to GCSE Maths lessons