National 5 Maths
Pythagoras' Theorem
Course content
- Pythagoras in complex situations
- Converse of Pythagoras' Theorem
- Pythagoras' Theorem extended to 3D
- Pythagoras in a circle.
Textbook page references
- Zeta National 5+ Maths
pp.189-198 - TeeJay Maths Book N5 pp.44-49
- Leckie National 5 Maths pp.205-217 and 226-233
Key ideas
- In any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the two perpendicular sides.
- In the diagram above, \(c\) is the hypotenuse (the longest side). So \(c^2=a^2+b^2\).
- If you are finding one of the shorter sides, \(a\) or \(b\), rearrange this equation and subtract.
N5 Maths revision course
National5.com self-study courseSave £10 with discount code 'Maths.scot'
Example 1 (non-calculator)
The diagonal of a square measures exactly 10 cm. Calculate the area of the square.
Example 2 (calculator)
A triangle has sides of length 28 cm, 45 cm and 52 cm.
Are the two shorter sides perpendicular? Justify your answer.
Example 3 (calculator)
The distances, as the crow flies, between three towns Ayton (A), Beeville (B) and Ceebury (C) are shown on the map, which is not drawn to scale.
Ayton is due east of Beeville.
Is Ceebury due north of Beeville? Give a reason for your answer.
Recommended student book
Zeta Maths: National 5 Maths Textbook
Best price, direct from the publisher
Example 4 (calculator)
SQA National 5 Maths 2023 P2 Q8
A wooden beam is used to support a wall built on horizontal ground as shown in the diagram.
The edge of the beam, AB, is 8 metres long.
C is at the foot of the wall.
A is 7 metres from C.
B is 4 metres from C.
Determine whether the wall is perpendicular to the ground.
Justify your answer.
Example 5 (calculator)
The cuboid ABCDEFGH has length 12 cm, breadth 7 cm and height 5 cm.
Calculate the length of its space diagonal AH, correct to 3 significant figures.
Recommended revision guides
How to Pass National 5 MathsBrightRED N5 Maths Study Guide
Example 6 (calculator)
SQA National 5 Maths 2018 P2 Q16
Chris wants to store his umbrella in a locker. The locker is a cuboid with internal dimensions of length 40 centimetres, breadth 40 centimetres and height 70 centimetres.
The umbrella is 85 centimetres long.
He thinks it will fit into the locker from corner P to corner M.
Is he correct?
Justify your answer.
Example 7 (non-calculator)
SQA National 5 Maths 2014 P1 Q12
The diagram below shows a circle, centre C.
The radius of the circle is 15 centimetres.
A is the mid-point of chord PQ.
The length of AB is 27 centimetres.
Calculate the length of PQ.
N5 Maths practice papers
Non-calculator papers and solutionsCalculator papers and solutions
Example 8 (calculator)
SQA National 5 Maths 2015 P2 Q12
The diagram below shows the circular cross-section of a milk tank.
The radius of the circle, centre O, is 1.2 metres.
The width of the surface of the milk in the tank, represented by ML in the diagram, is 1.8 metres.
Calculate the depth of the milk in the tank.
Example 9 (calculator)
SQA National 5 Maths 2017 P2 Q13
Two identical shapes are used to form a logo.
Each shape is part of a circle.
• The circles have centres C1 and C2.
• The radius of each circle is 14 centimetres.
• The logo has half-turn symmetry about the mid-point of AB.
• AB is 48 centimetres long.
Calculate the height of the logo.
Example 10 (calculator)
SQA National 5 Maths 2018 P2 Q12
The shape below is part of a circle, centre O.
The circle has radius 13 centimetres.
AB is a chord of length 20 centimetres.
Calculate the width of the shape.
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Maths.scot worksheet
| • Pythagoras' Theorem worksheet • Answer sheet • See all National 5 Maths worksheets |
Past paper questions
| • All past paper questions by topic |
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Pythagoras in a circle: • 2014 Paper 1 Q12 • 2015 Paper 2 Q12 • 2016 Paper 2 Q15 • 2017 Specimen Paper 1 Q16 • 2017 Paper 2 Q13 • 2018 Paper 2 Q12 • 2019 Paper 2 Q18 • 2021 Paper 1 Q18 • 2022 Paper 2 Q8 • 2023 Paper 1 Q10 • 2024 Paper 2 Q10 |
| Converse of Pythagoras: • 2014 Paper 2 Q6 • 2017 Paper 2 Q7 • 2019 Paper 2 Q11 • 2023 Paper 2 Q8 |
| 3D Pythagoras: • 2016 P1 Q7 (with 3D coordinates) • 2018 Paper 2 Q16 • 2022 Paper 2 Q11 |
Other great resources
| Videos - Mr Graham Maths 1. Nat 4 Pythagoras 2. Pythagoras in disguise 3. Pythagoras: repeated use 4. Distance between points 5. Converse of Pythagoras |
| Videos - St Benedict's Numeracy 1. Pythagoras basics 2. Converse of Pythagoras |
| PowerPoint - MathsRevision.com |
| Revision notes - BBC Bitesize Test yourself - BBC Bitesize |
| Lesson notes - Maths 777 |
| Worked examples - Maths Mutt |
| Practice questions - Maths Hunter |
| Worksheet - Brannock HS Pythagoras in circles (Answers) |
| Exercises - Larkhall Academy Pages 44-51 Ex 1-5 |
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