Angles in polygons Quiz (Grade 4)

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Topic:

Score: 0 / 10

Angles in polygons — Notes & Worked Examples Show / Hide

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

One Page Guide (PDF) Angles in polygons Worksheets

Question 1

[1 mark]

Calculate the size of an interior angle of a regular dodecagon

Worked Solution & Answer

Sum of interior angles of an n-sided polygon = (n - 2) × 180°
150°

Answer: 150° °

Question 2

[1 mark]

Calculate the size of an exterior angle of a regular nonagon

Worked Solution & Answer

Exterior angles of any polygon add up to 360°.
Interior + Exterior = 180°
40°

Answer: 40° °

Question 3 AQA Nov 2021 3F Q31 / 3H Q12

[2 marks]

Part of a regular polygon with 15 sides is shown.

Work out the size of an interior angle.

Answer degrees

Worked Solution & Answer

Exterior angle = 360°15 = 24° M1
Interior angle = 180° - 24° = 156° (or (15 - 2) × 180°15 = 156°). A1

Answer: Answer 156 degrees

Question Paper (PDF) • Mark Scheme (PDF)

Question 4 AQA Jun 2025 2H Q11

[3 marks]

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

Work out the size of angle x.

x =

Worked Solution & Answer

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

Question Paper (PDF) • Mark Scheme (PDF)

Question 5

[3 marks]

Here is a regular hexagon and a regular pentagon.

Work out the size of the angle marked x.
You must show all your working.

Worked Solution & Answer

Exterior angle of hexagon = 360 ÷ 6 = 60° M1
exterior angle of pentagon = 360 ÷ 5 = 72° M1
x = 60 + 72 = 132° (or 360 – 120 – 108 = 132°) A1

Answer: 132

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Question 6 AQA Nov 2020 2H Q20

[3 marks]

AB, BC, CD and DE are four of the sides of a regular decagon.

Work out the size of angle w.

Answer degrees

Worked Solution & Answer

Interior angle of regular decagon = (10 - 2) × 180°10 = 144° M1
In pentagon ABCDE, interior angles at B, C and D are each 144°
Sum of angles in a pentagon = (5 - 2) × 180° = 540° M1
By symmetry, ∠ EAB = ∠ DEA = w.
2w + 3(144°) = 540° ⇒ 2w + 432° = 540° ⇒ 2w = 108° ⇒ w = 54°. A1

Answer: Answer 54 degrees

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Question 7 IGCSE Jun 2023 2F Q27 / 2H Q11

[3 marks]

Here is a 9-sided regular polygon ABCDEFGHJ, with centre O

ODK and FEK are straight lines.
Work out the value of x

x =

Worked Solution & Answer

Exterior angle DEK = 360 ÷ 9 = 40° M1
Interior angle = 140° M1
Since OD bisects interior angle CDE, angle ODE = 70°, so angle EDK = 180 – 70 = 110°
In ∆DEK, x = 180 – 110 – 40 = 30. A1

Answer: x = 30

Question Paper (PDF) • Mark Scheme (PDF)

Question 8 AQA Jun 2023 2H Q6

[3 marks]

(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.

Answer

(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 & Answer

(a)

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

Answer (a): Answer 45

(b)

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)

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Question 9

[4 marks]

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.

n =

Worked Solution & Answer

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

Question Paper (PDF) • Mark Scheme (PDF)

Question 10

[4 marks]

In the diagram, AB, BC and CD are three sides of a regular polygon **P**.

Show that polygon **P** is a hexagon.
You must show your working.

Worked Solution & Answer

Interior angle of regular 12-sided polygon = 180 – 360/12 = 150° M1
Angles around point B sum to 360°, so interior angle of polygon P = 360 – 150 – 90 = 120° M1
Exterior angle of polygon P = 180 – 120 = 60° M1
Number of sides = 360 ÷ 60 = 6, so P is a hexagon. C1

Answer: Exterior angle of 12-gon = 30°, so interior angle = 150°. Angle at B in polygon P = 360° – 150° – 90° = 120°. Exterior angle of P = 180° – 120° = 60°, so number of sides = 360° ÷ 60° = 6 (hexagon).

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Score: 0 / 10 (0 / 27 marks)

Angles in polygons Worksheets Next: Surface area →

Grade 4 Quizzes

24 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