Pythagoras' Theorem: Complete Guide
Pythagoras' Theorem (a² + b² = c²) lets you find a missing side in any right-angled triangle. Here's how to spot when to use it, work through it without slipping up, and pick up the marks in the exam.
What is Pythagoras' Theorem?
Pythagoras' Theorem describes the relationship between the three sides of a right-angled triangle:
where c is the hypotenuse (the longest side, opposite the right angle)
In words: "The square of the hypotenuse equals the sum of the squares of the other two sides."
The hypotenuse is always:
- The LONGEST side
- Opposite the right angle (90°)
Finding the Hypotenuse
When you need to find the longest side (hypotenuse), you ADD the squares and square root.
Worked Example 1
A right-angled triangle has sides 3 cm and 4 cm. Find the hypotenuse.
Finding a Shorter Side
When you need to find one of the shorter sides, you SUBTRACT the squares and square root.
Rearranged to find a shorter side
Worked Example 2
A right-angled triangle has hypotenuse 13 cm and one side 5 cm. Find the other side.
Common Pythagorean Triples
These are sets of whole numbers that work in Pythagoras' Theorem. Memorise them to speed up your calculations!
- 3, 4, 5 (and multiples: 6, 8, 10 etc.)
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
- Don't forget to square root at the end! Students often stop at c² = 25 instead of c = 5
- Check which side is the hypotenuse - it must be the longest side
- Only works for right-angled triangles - look for the 90° angle symbol
When to Use Pythagoras
Use Pythagoras' Theorem when:
- You have a right-angled triangle
- You know TWO sides and need to find the THIRD
- You need to find distances (diagonal across a rectangle, distance between coordinates)