In a right-angled triangle, the square on the longest side equals the sum of the squares on the other two. Indian mathematicians wrote this down in the Baudhayana Sulbasutra centuries before Pythagoras.
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The hypotenuse is the side opposite the right angle, and it is always the longest.
📐 Using it in three steps
Find the right angle. The side opposite it is the hypotenuse, c.
Write a² + b² = c² with the two shorter sides as a and b.
Substitute and solve. To find a shorter side, rearrange: a² = c² − b².
The Indian name comes first for a reason. The Baudhayana Sulbasutra, written around 800 BCE, states the rule for building altars — well before Pythagoras. NCERT names both.
It only works for right-angled triangles. If there is no right angle, a² + b² = c² is simply false. Check for the right angle before you start.
Some whole-number trios fit the theorem exactly. Learning them saves you the square-root working.
🔢 Triples worth memorising
| a | b | c | Check |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 |
| 5 | 12 | 13 | 25 + 144 = 169 |
| 6 | 8 | 10 | 36 + 64 = 100 |
| 7 | 24 | 25 | 49 + 576 = 625 |
| 8 | 15 | 17 | 64 + 225 = 289 |
| 9 | 12 | 15 | 81 + 144 = 225 |
Multiples of a triple are also triples. From 3-4-5 you get 6-8-10, 9-12-15 and 30-40-50. If the numbers in a question look like a familiar triple scaled up, you can write the answer down immediately.
Ladders, shadows, diagonals and shortest paths are all the same triangle in disguise.
The classic ladder question: a ladder 13 m long leans against a wall with its foot 5 m away. How high does it reach? The ladder is the hypotenuse, so height² = 13² − 5² = 169 − 25 = 144, giving 12 m.
The diagonal of a rectangle splits it into two right triangles. For a 6 × 8 rectangle the diagonal is √(36+64) = √100 = 10. For a square of side a it is always a√2.
12 questions on right triangles. Find the hypotenuse first — it is opposite the right angle. 🍀
