Pythagoras' theorem states that in any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c², where c is always the side opposite the right angle. This one relationship lets you find any unknown side of a right-angled triangle if you know the other two sides.
What is Pythagoras' theorem?
a² + b² = c²
aandbare the two shorter sides (called the legs of the triangle).cis the hypotenuse — the longest side, always opposite the right angle.
The theorem is named after the ancient Greek mathematician Pythagoras (c. 570–495 BC), but evidence shows Babylonian and Indian mathematicians knew the relationship centuries earlier.
A visual proof
Draw a right-angled triangle with legs 3 cm and 4 cm. Build a square on each side:
- Square on leg
a = 3: area = 9 cm² - Square on leg
b = 4: area = 16 cm² - Square on hypotenuse
c: area = 9 + 16 = 25 cm², soc = 5 cm
This 3-4-5 triangle is the most famous Pythagorean triple — a set of three whole numbers that satisfy a² + b² = c².
Step-by-step method
Finding the hypotenuse (the longest side)
- Identify the right angle and label the hypotenuse
c. - Label the two shorter sides
aandb. - Substitute into
a² + b² = c². - Calculate
a² + b². - Square-root the result to find
c.
Finding a shorter side
- Identify which shorter side is unknown.
- Rearrange:
a² = c² − b²(orb² = c² − a²). - Substitute the known values.
- Square-root to find the missing side.
Worked examples
Worked example 1 — finding the hypotenuse
A right-angled triangle has legs 6 cm and 8 cm. Find the hypotenuse.
a² + b² = c²
6² + 8² = c²
36 + 64 = c²
100 = c²
c = √100 = 10
Answer: 10 cm
Check: 6-8-10 is a multiple of the 3-4-5 triple (multiply each by 2).
Worked example 2 — finding the hypotenuse (non-integer answer)
A right-angled triangle has legs 5 cm and 7 cm. Find the hypotenuse, giving your answer to 1 decimal place.
5² + 7² = c²
25 + 49 = c²
74 = c²
c = √74 ≈ 8.6
Answer: 8.6 cm (1 d.p.)
Worked example 3 — finding a shorter side
A right-angled triangle has hypotenuse 13 cm and one leg 5 cm. Find the other leg.
a² + b² = c²
a² + 5² = 13²
a² + 25 = 169
a² = 169 − 25 = 144
a = √144 = 12
Answer: 12 cm
Check: 5-12-13 is a Pythagorean triple.
Worked example 4 — finding a shorter side (non-integer answer)
A right-angled triangle has hypotenuse 10 cm and one leg 4 cm. Find the other leg to 2 decimal places.
b² = c² − a²
b² = 10² − 4²
b² = 100 − 16 = 84
b = √84 ≈ 9.17
Answer: 9.17 cm (2 d.p.)
Worked example 5 — real-world context (ladder problem)
A ladder 5 m long leans against a vertical wall. The foot of the ladder is 2 m from the base of the wall. How far up the wall does the ladder reach? Give your answer to 2 decimal places.
The wall is vertical (90° to the ground), so the triangle is right-angled.
- Hypotenuse = 5 m (the ladder)
- One leg = 2 m (horizontal distance from wall)
- Unknown leg = height up the wall
a² = c² − b²
a² = 5² − 2²
a² = 25 − 4 = 21
a = √21 ≈ 4.58
Answer: 4.58 m (2 d.p.)
Worked example 6 — testing whether a triangle is right-angled
Is a triangle with sides 7 cm, 24 cm, and 25 cm right-angled?
Test: does 7² + 24² = 25²?
49 + 576 = 625
625 = 625 ✓
Yes, the triangle is right-angled. (This is a Pythagorean triple.)
Common Pythagorean triples
These sets of whole numbers satisfy a² + b² = c². Recognising them saves time in exams.
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
Any multiple of a triple is also a triple: 6-8-10 (× 2 of 3-4-5), 9-12-15 (× 3 of 3-4-5), and so on.
How to identify the hypotenuse correctly
The hypotenuse is always opposite the right angle — not necessarily the side with the biggest number you are told, and not a side adjacent to the right angle. In a diagram, the right angle is marked with a small square. The side directly across from that square is the hypotenuse.
How Pythagoras' theorem fits the KS3 curriculum
The Department for Education's KS3 mathematics programme of study requires pupils to "apply angle facts, triangle congruence, similarity, and properties of quadrilaterals to conjecture and derive results about angles and sides, including the Pythagorean theorem." BBC Bitesize KS3 geometry covers Pythagoras' theorem with interactive diagrams and practice questions for Year 9 pupils. The theorem reappears throughout GCSE — in trigonometry, 3D problems, and coordinate geometry — making a solid KS3 foundation essential.
Common mistakes
Mistake 1 — Adding the squares when you should subtract.
Finding the hypotenuse: a² + b² = c² (add). Finding a shorter side: a² = c² − b² (subtract). Students often add even when looking for a shorter side.
Mistake 2 — Square-rooting each term individually.
√(a² + b²) is NOT a + b. For example, √(9 + 16) = √25 = 5, not 3 + 4 = 7. You must add the squares first, then take the square root of the total.
Mistake 3 — Not identifying the hypotenuse correctly.
The hypotenuse is the side opposite the right angle — it is always c in the formula. Using a leg as c gives a completely wrong answer.
Mistake 4 — Forgetting to square-root at the end.
After calculating c² = 100, the answer is c = 10, not c = 100. Always square-root to get the side length.
Frequently asked questions
What does a² + b² = c² actually mean?
It means that if you build a square on each side of a right-angled triangle, the areas of the two smaller squares added together equal the area of the square on the hypotenuse. So for a 3-4-5 triangle: a square of area 9 plus a square of area 16 gives a square of area 25. It is a statement about areas, not just lengths.
Does Pythagoras' theorem work for all triangles?
No — only for right-angled triangles (triangles with one angle of exactly 90°). For other triangles you would use the cosine rule at GCSE, which generalises Pythagoras' theorem. You can use Pythagoras' theorem to test whether a triangle is right-angled: if a² + b² = c² is satisfied with the longest side as c, the triangle has a right angle.
How do I know whether to add or subtract the squares?
Ask yourself: am I finding the hypotenuse (longest side) or a shorter side? If the hypotenuse is unknown, add the squares of the two known sides, then square-root. If a shorter side is unknown, subtract the square of the known shorter side from the square of the hypotenuse, then square-root.
Why does Pythagoras' theorem appear in so many GCSE topics?
Right-angled triangles appear everywhere: in 2D geometry (diagonals of rectangles, height of isosceles triangles), coordinate geometry (distance between two points uses Pythagoras), trigonometry (sin, cos, tan are defined using right-angled triangles), and 3D problems (finding the diagonal of a cuboid). Pythagoras' theorem is the tool that connects these topics, which is why mastering it at KS3 is so valuable.
For Socratic KS3 geometry tutoring on Pythagoras' theorem, visit aitutors.me.