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GCSE Maths · Geometry and measures

Angles in polygons GCSE Questions and Worked Answers

A simple polygon with n sides has interior-angle sum (n − 2) × 180°. A regular polygon has equal angles; each exterior angle is 360° ÷ n and each interior angle is 180° minus that.

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Start with the meaning

What you need to know about angles in polygons

A triangle is a closed shape made from three straight sides. A quadrilateral has four sides, a pentagon five and a hexagon six. These are polygons. Their interior angles are the angles inside the boundary at each corner.

See the idea first

Split the shape into triangles

A triangle's angles total 180°. A diagonal joins two non-neighbouring corners. In a convex polygon, diagonals from one corner divide it into triangles without crossing outside. A hexagon produces four triangles, so its interior angles total 720°. In general, n sides produce n − 2 triangles.

A
A six-sided polygon splits into four triangles. None of these diagonals adds a new corner, so its interior angles total 4 × 180°.
From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Find an angle sum

What the problem asks: Find the sum of the interior angles of an octagon.

How to solve it: Eight sides produce 8 − 2 = 6 triangles, so the sum is 6 × 180° = 1080°. The octagon does not need to be regular.

Find one angle of a regular polygon

What the problem asks: Find each interior angle of a regular pentagon.

How to solve it: Its sum is 3 × 180° = 540°. Regular means equal sides and equal angles, so divide by five: 108°.

Find the number of sides

What the problem asks: A regular polygon has an exterior angle of 24°. How many sides?

How to solve it: Walking once around a convex polygon turns you through 360° in total. Equal 24° turns require 360 ÷ 24 = 15 sides.

Find an irregular missing angle

What the problem asks: A quadrilateral has interior angles 75°, 110°, 95° and x. Find x.

How to solve it: The sum is 360°, whether regular or not. Subtract known angles: x = 360 − 75 − 110 − 95 = 80°.

A reliable routine

Choose the angle relationship for the actual task

Interior-angle sums come from triangles. Exterior angles are the turns made as you follow the edges around a convex polygon: one complete journey turns through 360°. An interior angle and its adjacent exterior angle form a straight angle, so they total 180°.

  1. Identify whether the question gives an interior angle, exterior angle or total sum.
  2. Use (n − 2) × 180° for an interior sum, with n the number of sides.
  3. Divide an interior sum by n only when all angles are equal.
  4. For a regular polygon use exterior = 360° ÷ n; check that a proposed side count is an integer of at least 3.

Check: The ordinary angle-sum formula applies to simple, non-self-intersecting polygons. The one-corner fan diagram here assumes a convex polygon; a concave one may need a different triangulation.

Fully worked

Angles in polygons GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Sum for seven sides

2 marks
Question

Find the sum of the interior angles of a heptagon.

A heptagon has n = 7 sides.

S=(7−2)180∘S=(7-2)180^\circ =5(180∘)=900∘=5(180^\circ)=900^\circ

Regularity is not needed for this sum.

Example 2

Regular interior angle

3 marks
Question

Find each interior angle of a regular octagon.

S=(8−2)180∘=1080∘S=(8-2)180^\circ=1080^\circ

Equal angles mean each is

1080∘/8=135∘1080^\circ/8=135^\circ

Tip: do not divide by n unless the angles are equal.

Example 3

Exterior angle

2 marks
Question

Find each exterior angle of a regular nonagon.

A nonagon has nine sides and equal exterior turns.

e=360∘/9=40∘e=360^\circ/9=40^\circ

The adjacent interior angle is 180° − 40° = 140°.

Example 4

Sides from an interior angle

3 marks
Question

Each interior angle of a regular polygon is 156°. Find its number of sides.

First find the exterior turn.

e=180∘−156∘=24∘e=180^\circ-156^\circ=24^\circ n=360∘/24∘=15n=360^\circ/24^\circ=15

There are 15 sides.

Example 5

Missing irregular angle

3 marks
Question

A pentagon has interior angles 90°, 115°, 100°, 125° and x. Find x.

S=(5−2)180∘=540∘S=(5-2)180^\circ=540^\circ x=540∘−(90∘+115∘+100∘+125∘)x=540^\circ-(90^\circ+115^\circ+100^\circ+125^\circ) x=540∘−430∘=110∘x=540^\circ-430^\circ=110^\circ

The angles need not be equal.

Example 6

Angles meeting at a point

3 marks
Question

A regular hexagon and a square share a common side and lie on opposite sides of it. At one end of that side, find the angle of the gap between the two shapes.

Hexagon interior angle: (6−2)180∘/6=120∘(6-2)180^\circ/6=120^\circ. Square angle: 90°. A complete turn is 360°.

x=360∘−120∘−90∘=150∘x=360^\circ-120^\circ-90^\circ=150^\circ

Use interior angles where the shapes meet.

10 original questions · total 24 marks

Angles in polygons GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 29 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
1

Pentagon sum

2 marks

Find the interior-angle sum of a pentagon.

Show worked answer
S=(5−2)180∘=540∘S=(5-2)180^\circ=540^\circ
2

Decagon sum

2 marks

Find the interior-angle sum of a decagon.

Show worked answer

Ten sides give eight triangles:

S=8(180∘)=1440∘S=8(180^\circ)=1440^\circ
3

Regular hexagon

2 marks

Find one interior angle of a regular hexagon.

Show worked answer
S=(6−2)180∘=720∘S=(6-2)180^\circ=720^\circ 720∘/6=120∘720^\circ/6=120^\circ
4

Regular pentagon exterior

2 marks

Find each exterior angle of a regular pentagon.

Show worked answer
e=360∘/5=72∘e=360^\circ/5=72^\circ

Five equal turns make one full turn.

5

From an exterior angle

2 marks

A regular polygon has exterior angle 18°. Find its number of sides.

Show worked answer
n=360∘/18∘=20n=360^\circ/18^\circ=20
6

From an interior angle

3 marks

A regular polygon has interior angle 165°. Find its number of sides.

Show worked answer
e=180∘−165∘=15∘e=180^\circ-165^\circ=15^\circ n=360∘/15∘=24n=360^\circ/15^\circ=24
7

Irregular quadrilateral

2 marks

A quadrilateral has angles 88°, 97°, 105° and x. Find x.

Show worked answer
x=360∘−88∘−97∘−105∘=70∘x=360^\circ-88^\circ-97^\circ-105^\circ=70^\circ
8

From the sum

3 marks

A polygon has interior-angle sum 1260°. Find the number of sides.

Show worked answer
(n−2)180=1260(n-2)180=1260 n−2=7n-2=7 n=9n=9

It is a nonagon; it need not be regular.

9

Is it possible?

3 marks

Can a regular polygon have each exterior angle 25°? Explain.

Show worked answer
360/25=14.4360/25=14.4

A polygon cannot have 14.4 sides. Therefore no regular polygon has this exterior angle.

10

An algebraic quadrilateral

3 marks

A quadrilateral's angles are x°, 2x°, 3x° and 4x°. Find all four angles.

Show worked answer
x+2x+3x+4x=360x+2x+3x+4x=360 10x=36010x=360 x=36x=36

Angles are 36°, 72°, 108°, 144°. Their sum checks to 360°.

Examiner-style feedback

Common angles in polygons mistakes

Using 360° for every interior sum

360° is a quadrilateral's interior sum, or a complete set of exterior turns of a convex polygon.

Confusing sum with each angle

A total must be divided by the number of equal angles to find one regular interior angle.

Dividing by the interior angle

To recover n with 360 ÷ angle, use the exterior turn, not the interior angle.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

  1. Count sides.
  2. Separate interior angles from exterior turns.
  3. Use triangle sums or one full turn.
  4. Use equal-angle division only when justified.
Quick answers

Angles in polygons FAQ

What does regular mean?

All sides have equal length and all interior angles are equal.

Do irregular polygons use the same angle sum?

Yes, if they are simple polygons. Their individual angles need not be equal.

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What to revise next

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Content standards

Curriculum and rights review

G1/G3/G4 polygon vocabulary, sums and regular angle problems across tiers. Diagrams and turn arguments use convex polygons unless noted. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references