GCSE MATHS • HIGHER TIER
Basic Pythagoras Made Simple
Find missing sides in right-angled triangles using Pythagoras’ Theorem accurately and confidently.
Clear method
Exam practice
Worked proofs
Basic Pythagoras in Four Moves
1
Identify
Check the triangle is right-angled and identify the hypotenuse.
2
Substitute
Write a² + b² = c² and substitute the known side lengths.
3
Solve
Calculate the squares, rearrange if needed and square root the result.
4
Check
Check the missing side is sensible and include the correct units.
What You’ll Learn
Understand Pythagoras' Theorem.
Identify the hypotenuse correctly.
Find a missing hypotenuse.
Find a missing shorter side.
Use a clear step-by-step method.
Apply Pythagoras to real-life distance problems.
Key Rules
Pythagoras’ Theorem applies only to right-angled triangles.
The hypotenuse is the longest side and is opposite the right angle.
Use the formula :
a2+b2=c2
c always represents the hypotenuse.
Worked Proof: Basic Pythagoras
Step-by-Step Method
- 1. Check that the triangle is right-angled.
- 2. Identify and label the hypotenuse.
- 3. Write \(a^2 + b^2 = c^2\).
- 4. Substitute the known side lengths.
- 5. Add the squares for the hypotenuse or subtract them for a shorter side.
More Worked Examples
1
Find the hypotenuse when the other sides are 3 cm and 4 cm.
Step 1: Write the formula.
a² + b² = c²
Step 2: Substitute values.
3² + 4² = c²
Step 3: Calculate squares.
9 + 16 = 25
Step 4: Square root.
c = 5
Step 2: Substitute values.
3² + 4² = c²
Step 3: Calculate squares.
9 + 16 = 25
Step 4: Square root.
c = 5
2
Find the hypotenuse when sides are 5 cm and 12 cm.
5² + 12² = c²
25 + 144 = 169
c = 13
c = 13
3
Hypotenuse = 13 cm and one side = 5 cm.
5² + b² = 13²
25 + b² = 169
b² = 144
b = 12
b² = 144
b = 12
4
Hypotenuse = 10 cm and one side = 6 cm.
6² + b² = 10²
36 + b² = 100
b² = 64
b = 8
b² = 64
b = 8
- Common GCSE Mistakes
- Using the formula on non-right-angled triangles.
- Choosing the wrong side as the hypotenuse.
- Forgetting to square numbers.
- Forgetting to square root the final answer.
- Making arithmetic errors.
- Exam Tips
- Look for the right-angle symbol before using Pythagoras.
- Identify the hypotenuse before substituting any values.
- Use addition when finding the hypotenuse.
- Rearrange and subtract when finding a shorter side.
- Keep calculator values accurate until the final answer.
- Check whether the missing side should be the longest or a shorter side.
Skills Used in Basic Pythagoras
• Pythagoras' Theorem
Right-Angled Triangles
Squares
Square Roots
Rearranging Formulae
Geometry
Estimation
Problem Solving
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Exam-Style Questions
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Frequently Asked Questions
In a right-angled triangle, the square of the longest side equals the sum of the squares of the other two sides.
The hypotenuse is the longest side of a right-angled triangle and is always opposite the right angle.
a² + b² = c², where c is the hypotenuse.
Use it when the triangle is right-angled, two sides are known and one side is missing.
Use the known hypotenuse and subtract the square of the known shorter side before taking the square root.
The source recommends SOHCAHTOA, Exact Trig Values, Sine Rule, Cosine Rule, Trigonometry in 3D and Bearings.