Pythagoras’ Theorem
One of the most famous formulas in all of maths. It lets you find a missing side of any right-angled triangle — and it comes up in loads of exam questions.
The Formula
a² + b² = c²
c is always the hypotenuse — the longest side, directly opposite the right angle. aand b are the other two sides (order doesn’t matter).
Finding the Long Side (Hypotenuse)
If you know the two shorter sides, square them both, add them together, then square root the result.
Example 1
Sides: 3 cm and 4 cm
c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5 cm
Example 2
Sides: 6 cm and 8 cm
c² = 36 + 64 = 100
c = √100 = 10 cm
Example 3
Sides: 5 cm and 9 cm
c² = 25 + 81 = 106
c = √106 = 10.30 cm (2 d.p.)
Finding a Short Side
Rearrange: a² = c² − b². Square the hypotenuse, subtract the other known side squared, then square root.
Example 1
Hypotenuse = 13 cm, one side = 5 cm
a² = 13² − 5² = 169 − 25 = 144
a = √144 = 12 cm
Example 2
Hypotenuse = 10 cm, one side = 7 cm
a² = 100 − 49 = 51
a = √51 = 7.14 cm (2 d.p.)
Real-World Uses
- Ladder against a wall: A 5 m ladder leans against a wall with its base 3 m from the wall. How high up does it reach? √(25 − 9) = √16 = 4 m.
- TV screen diagonal: A TV is 80 cm wide and 60 cm tall. The diagonal = √(6400 + 3600) = √10000 = 100 cm.
Pythagorean Triples
Some sets of whole numbers fit Pythagoras perfectly. Knowing them saves time (and impresses examiners):
- 3, 4, 5 (and multiples: 6,8,10 — 9,12,15 etc.)
- 5, 12, 13
- 8, 15, 17
Common Mistakes
- Adding instead of subtracting when finding a shorter side.
- Forgetting to square root at the end (you find c², not c).
- Using Pythagoras on a triangle that isn’t right-angled.
- Labelling the wrong side as the hypotenuse.
Practice Questions
- Find the hypotenuse: sides 9 cm and 12 cm.
- Find the hypotenuse: sides 7 cm and 24 cm.
- Find the missing side: hypotenuse 17 cm, one side 8 cm.
- A rectangle is 12 cm by 5 cm. Find the diagonal.
- A ship sails 8 km east then 6 km north. How far is it from the start in a straight line?
Answers: 1) 15 cm 2) 25 cm 3) 15 cm 4) 13 cm 5) 10 km