National 5 Maths
Angles in Shapes

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Course 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.
×

Quadrilateral

A two-dimensional shape with four edges.

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

×

Polygon

A two-dimensional shape with straight edges.

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

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

Textbook page references

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.

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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.

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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.

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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
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

Other great resources

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
Lesson notes - Maths 777
1. Angles in polygons
2. Angles in circles
3. Pythagoras and trig in circles
Worked examples - Maths Mutt
Revision notes - BBC Bitesize
1. Triangles
2. Quadrilaterals
3. Polygons
4. Circles
Worksheet - Brannock HS
Pythagoras in circles (Answers)
Exercises - Larkhall Academy
Pages 1-21 Ex 1-5 (no answers)

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