Class 10 · Maths · Triangles

Pythagoras Theorem

4 min readUpdated November 2026

In one line

In a right-angled triangle, the square of the longest side equals the sum of the squares of the other two sides.

The formula

a² + b² = c²

The hypotenuse c is always opposite the right angle and is the longest side.

Key points

It only works for right-angled triangles (one angle is exactly 90°).
The hypotenuse is the side opposite the right angle. It's always the longest side.
To find the hypotenuse: c = √(a² + b²).
To find a shorter side: a = √(c² − b²).
Converse: if the square of one side equals the sum of the squares of the other two, the triangle is right-angled, and the right angle is opposite the longest side.
Pythagorean triplets are whole numbers that fit the rule: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25). Multiples work too, like (6, 8, 10).

Solved examples

1. Find the hypotenuseThe two shorter sides are 9 cm and 12 cm.
c² = 9² + 12² = 81 + 144 = 225, so c = √225 = 15 cm.

2. The ladder problemA 10 m ladder leans on a wall. Its foot is 6 m from the wall. How high does it reach?
The ladder is the hypotenuse. Height² = 10² − 6² = 100 − 36 = 64, so height = 8 m.

3. Is it a right triangle?Sides 7 cm, 24 cm and 25 cm.
7² + 24² = 49 + 576 = 625 = 25². Yes, it is right-angled, with the right angle opposite the 25 cm side.

Common mistakes

Using it on any triangle. Check for a 90° angle first.
Putting the wrong side as c. c must be the side opposite the right angle.
Forgetting the square root. c² = 225 means c = 15, not 225.
Adding when you should subtract. If you know the hypotenuse and want a shorter side, subtract: a² = c² − b².

Quick check

Revise in 30 seconds

Rule	a² + b² = c² (right triangles only)
Hypotenuse	c = √(a² + b²)
Shorter side	a = √(c² − b²)
Converse	If a² + b² = c², the triangle is right-angled
Triplets	(3,4,5) · (5,12,13) · (8,15,17) · (7,24,25)