Where can I download free angles in polygons PDF worksheets with answers?

On this MathsLinks page you can download free printable angles in polygons PDF worksheets (Qs) and full worked solutions or mark schemes (Ans) from Ken Stafford, Ken Stafford (1), Maths Genie, Maths4Everyone, Corbett Practice, plus a printable one-page angles in polygons revision guide.

What GCSE grade is angles in polygons (4M)?

Angles in polygons (topic code 4M) is a Grade 4 geometry topic in GCSE Maths for Edexcel, AQA, OCR, Eduqas and IGCSE specifications.

Free Angles in polygons Worksheets (PDF) with Answers — GCSE Maths

Free Angles in polygons PDF Worksheets & Answers

Grade 4 • Topic 4M

Resource	Questions	Answers
Ken Stafford GCSE past paper questions & solutions	Qs	Ans
Ken Stafford (1) Practice worksheet & worked solutions	Qs	Ans
Maths Genie Graded exam questions & solutions	Qs	Ans
Maths4Everyone GCSE 9–1 practice booklet & solutions	Qs	Ans
Corbett Practice Practice Questions & answers	Qs	Ans
Corbett Textbook Textbook Exercise & answers	Qs	Ans
Maths Made Easy MME worksheet & model answers	Qs	Ans
Dr Frost GCSE worksheet & solutions	Qs	Ans
Naiker Maths Exam question booklet & solutions	Qs	Ans
Dr Austin (1) Practice grid / worksheet & answers	Qs	Ans
Dr Austin (2) Practice grid / worksheet & answers	Qs	Ans
Dr Austin (3) Practice grid / worksheet & answers	Qs	Ans
1st Class Maths (Edxl) Edexcel exam questions & solutions	Qs	Ans
1st Class Maths (AQA) AQA exam questions & solutions	Qs	Ans
Wordsearch Angles in polygons questions & answers	Qs	Ans
Metatutor Topic worksheet & worked solutions	Qs	Ans
GCSE Maths Rocket Practice questions & answers	Qs	Ans
E Classroom GCSE topic booklet & answers	Qs	Ans
PMT (Edxl) Edexcel GCSE topic questions & solutions	Qs	Ans
PMT (AQA) AQA GCSE topic questions & solutions	Qs	Ans
PMT (IGCSE) Edexcel IGCSE topic questions & solutions	Qs	Ans
Cognito Interactive online lesson & practice	Page
MathsPad interactive resource Interactive online resource	Open

Grade 4 topics
4A - Compound interest	4B - Indices (index laws)
4C - Prime factors HCF and LCM	4D - Real life graphs
4E - Inequalities on a number line	4F - Solving inequalities
4G - Forming and solving equations	4H - nth term of a linear sequence
4J - Expanding and factorising	4K - Pythagoras theorem
4L - Angles in parallel lines	4M - Angles in polygons
4N - Surface area	4P - Volume of a cuboid
4Q - Volume of a prism	4R - Bearings
4S - Cylinders	4T - Plans and elevations
4U - Scatter graphs	4V - Probability
4W - Averages from tables	4X - Reverse mean
4Y - Area of a trapezium	4Z - Loci and construction

Explanation:

For any n-sided polygon: the Sum of interior angles = (n − 2) × 180°, and the Sum of exterior angles = 360°. At every vertex, the interior and exterior angles lie on a straight line, so Interior + Exterior = 180°.

For a regular polygon (all sides and angles equal), it is always fastest to use the exterior angle first: One Exterior Angle = 360°n, and Number of sides n = 360°One Exterior Angle.

Example 1: Work out one interior angle of a regular decagon (10 sides).

Exterior = 360° ÷ 10 = 36°

Interior = 180° − 36° = 144°

Example 2: A regular polygon has an interior angle of 165°. How many sides does the polygon have?

Solution:

Find the exterior angle first:

Exterior angle = 180° − 165° = 15°

Number of sides n = 360° ÷ 15° = 24

Answer: 24 sides

Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Angles in polygons Questions & Answers

10-Question Interactive Quiz More Past Paper Questions

Past Paper Example 1 GCSE May 2025 1F Q24 / 1H Q6 4 marks

Full Paper (PDF) • Mark Scheme (PDF)

AB, BC, CD and DE are four sides of a regular polygon with n sides.

Work out the value of n.
You must show all your working.

BCX is an equilateral triangle.
CDYX is a square.

Worked Solution & Mark Scheme (4 marks)

Equilateral triangle angle BCX = 60° M1
Square angle CDX = 90° M1
Interior angle of polygon at C = 60° + 90° = 150° M1
Exterior angle = 180° – 150° = 30°
Number of sides n = 360° ÷ 30° = 12. A1

Answer: n = 12

Past Paper Example 2 AQA Jun 2023 2H Q6 3 marks

Full Paper (PDF) • Mark Scheme (PDF)

(a) Assume the regular polygon is an octagon. Work out the size of an exterior angle. [2 marks]
6 (b) In fact, the polygon has more sides than an octagon. What does this mean about the size of an exterior angle? [1 mark]

(a) Part of a regular polygon is shown. [2 marks]

Assume that the polygon is an octagon.
Work out the size of an **exterior** angle.

(b) In fact, the polygon has **more** sides than an octagon.
What does this mean about the size of an exterior angle?
Tick **one** box.
[ ] It is more than the answer to part (a)
[ ] It is the same as the answer to part (a)
[ ] It is less than the answer to part (a)
[ ] It could be any of the above [1 mark]

Worked Solution & Mark Scheme (3 marks)

Part (a) Solution:

Exterior angle of regular octagon = 360° ÷ 8 = 45° M1 A1

Answer (a): Answer 45

Part (b) Solution:

Dividing 360° by a number greater than 8 gives an angle less than 45° B1

Answer (b): It is less than the answer to part (a)

Past Paper Example 3 AQA Jun 2025 2H Q11 3 marks

Full Paper (PDF) • Mark Scheme (PDF)

A straight line is drawn across a **regular** octagon.

Work out the size of angle x.

Worked Solution & Mark Scheme (3 marks)

Exterior angle of regular octagon = 360° ÷ 8 = 45° M1
Interior angle = 180° – 45° = 135° M1
By symmetry, the line bisects the interior angle: x = 135° ÷ 2 = 67.5°. A1

Answer: x = 67.5

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Angles in polygons

Free Angles in polygons PDF Worksheets & Answers

Guide to Angles in polygons: Explanation, Diagrams & Worked Examples

GCSE Past Paper Angles in polygons Questions & Answers