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National 5 Maths
Angles in Shapes

Page sections

Topic content

  • Applying the properties of shapes to determine an angle.
  • The working will involve at least two steps.
  • The shapes could be triangles, quadrilaterals , other polygons or circles.
  • Circle questions could involve the relationship between the centre, a chord and a perpendicular bisector.
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Quadrilateral

A closed two-dimensional shape with four edges.

Examples: square, rectangle, rhombus, parallelogram, trapezium, kite, etc.

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Polygon

A closed two-dimensional shape with straight edges.

Examples: triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon etc.

The word polygon isn't often used to refer to triangles or quadrilaterals . It usually implies at least five sides.

Polygons with large numbers of sides don't have special names. For example, a polygon with 38 sides is called a 38-gon.

If a polygon is regular, all of its edges have equal length.

Textbook page numbers

  • Zeta National 5+ Maths pp.181-188
  • TeeJay Maths N5 Book 4+ pp.139-146
  • Leckie National 5 Maths pp.218-237

Buy our favourite textbook

Zeta National 5+ Maths
Clear and comprehensive.
Progressive exercises.
Includes answers.
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Key ideas

  • The sum of the three angles in any triangle is 180°.
  • The sum of the four angles in any quadrilateral is 360°.
  • The sum of the exterior angles of any polygon is 360°.
  • The angle between a radius and the corresponding tangent to a circle is 90°.
  • When a triangle is formed inside a semicircle, the angle at the point not on the diameter is always 90°.

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Example 1 (non-calculator)

Calculate the size of each angle inside a regular octagon. [Note for ChatGPT: An accompanying diagram in this question shows the two different methods. Diagram 1 illustrates method 1, which a 135 degree interior angle and 45 degree exterior angle. Diagram 2 illustrates method 2, with a 45 degree central angle and two of the adjacent equal angles in two adjacent isosceles triangles adding to 135 degrees.]

Example 2 (non-calculator)

SQA National 5 Maths 2015 P1 Q3

AC is a tangent to the circle, centre O, with point of contact B.
DE is a diameter of the circle and F is a point on the circumference.
Angle ABD is 77° and angle DEF is 64°.
Calculate the size of angle BDF. [Note for ChatGPT: An accompanying diagram in this question shows ABC as a tangent to the circle. B is a point on the circumference. Point F is in one semicircle formed by the diameter DE and point B is in the other semicircle formed by the diameter DE. It can therefore be seen in the diagram that angle BDF equals angle BDO plus angle ODF.]

Example 3 (calculator)

SQA National 5 Maths 2016 P2 Q5

The diagram below shows a circle, centre O.

• AB and CB are tangents to the circle.
• AC and ED are parallel.
• Angle AOD is 143°.
Calculate the size of angle ABC. [Note for ChatGPT: An accompanying diagram in this question shows that DE is a diameter of the circle, as it passes through point O. Points A and C are on the circumference. Point B is outside the circle. Point B is the intersection point of tangents AB and CB.]

Example 4 (non-calculator)

SQA National 5 Maths 2017 P1 Q9

In the diagram shown below:
• ABE is a tangent to the circle centre O
• Angle DBE is 58°

Calculate the size of angle CAB. [Note for ChatGPT: An accompanying diagram in this question shows that B is a point on the circumference of the circle. CD can be seen to be a diameter as point O is on CD. Point A is outside the circle, and ACOD can be seen to be a straight line. CB, OB and DB are joined in the diagram, forming two triangles inside the circle.]

Example 5 (non-calculator)

SQA National 5 Maths 2022 P1 Q4

The diagram below shows a circle with centre O.

AB is a tangent to the circle at the point C.
CD is a diameter of the circle.
Angle EOD is 68°.
Calculate the size of angle ACE.

Example 6 (calculator)

SQA National 5 Maths 2023 P2 Q5

A logo consists of an H shape and a regular decagon.
The diagram represents the logo.

Calculate the size of the shaded angle. [Note for ChatGPT: An accompanying diagram in this question shows a decagon with ten dotted lines through its central point. The "H shape" comprises two vertical parallel lines and a horizontal transversal. One of the vertical bars in coincident with one of the sides of the decagon. The shaded angle comprises one exterior angle of the decagon and one of the right angles inside the "H" shape, added. Do not use the words "transversal" or "coincident" in your explanation.]

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Buy N5 Maths practice papers

Zeta: Five Practice Papers   TOP CHOICE
CGP: N5 Maths Exam Practice 
Leckie: Revision and Practice 
Hodder: N5 Maths Practice Papers 

Past paper questions

• All past paper questions by topic
Angles in polygons:
• 2018 Paper 1 Q9
• 2019 Paper 1 Q11 (also circle)
• 2023 Paper 2 Q5
• 2025 Paper 2 Q7
Angles in circles:
• 2015 Paper 1 Q3
• 2016 Paper 2 Q5
• 2017 Paper 1 Q9
• 2019 Paper 1 Q11 (also polygon)
• 2021 Paper 1 Q4
• 2022 Paper 1 Q4
• 2024 Paper 1 Q10
• 2026 Paper 1 Q10
Intermediate 2 (2000–2015)
• Exam questions (with answers)

Buy our favourite N5 textbook

Zeta National 5+ Maths
Clear and comprehensive.
Progressive exercises.
Includes answers.
Buy from Zeta Press 

Worksheets

Essential Skills worksheet
• Pythagoras in circles (Answers)
Corbettmaths worksheets
1. Angles in polygons (Answers)
2. Angles in circles (Answers)
National5Maths.co.uk worksheet
• Angles in a circle (no answers)
Maths4Everyone worksheet
• Isosceles in circle (with answers)
Larkhall Academy exercises
• Pages 1-21 Ex 1-5 (no answers)

Buy N5 Maths revision guides

How to Pass N5 Maths    TOP CHOICE
BrightRED: N5 Maths Study Guide 
CGP: N5 Maths Revision Guide 

Notes and videos

Videos - Larbert High School
1. Angles in polygons
2. Tangents to circles
3. Isosceles triangles in circles
Video - Mr Graham Maths
Angles in circles
PowerPoint - MathsRevision.com
Worked examples - Maths Mutt
Revision notes - BBC Bitesize
1. Triangles
2. Quadrilaterals
3. Polygons
4. Circles

All original content is © 2020– Andrew Moulden, unless otherwise indicated. Full copyright notice.
SQA / Qualifications Scotland material is copyright © Qualifications Scotland, used by kind permission.

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