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National 5 Maths
Angles in Shapes

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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

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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.

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.

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.

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.

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.

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
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

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