Pythagoras Theorem: Statement, Proof, Formula & Solved Examples

By LearnIQ · Updated 15 June 2026 · 7 min read

The Pythagoras theorem is one of the most important results in geometry and a favourite of board examiners. If you are in Class 9 or Class 10, mastering it unlocks a large chunk of marks in coordinate geometry, trigonometry, and mensuration. This guide explains what the theorem says, proves it from scratch, and walks through solved examples the way they appear in CBSE and state-board papers.

What does the Pythagoras theorem state?

In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The hypotenuse is always the longest side and lies directly across from the 90° angle.

If a triangle has legs of length a and b, and a hypotenuse of length c, then:

a² + b² = c²

This single relationship lets you find any one side of a right triangle when you know the other two.

A simple proof of the theorem

There are over 350 known proofs, but the area-based proof is the easiest to follow. Take four identical right-angled triangles with legs a and b and hypotenuse c. Arrange them inside a large square of side (a + b) so that their hypotenuses form a tilted square in the middle.

• The area of the large outer square is (a + b)².
• The same area also equals the four triangles plus the inner tilted square: 4 × (½ × a × b) + c².
• Setting them equal: (a + b)² = 2ab + c².
• Expanding the left side: a² + 2ab + b² = 2ab + c².
• Cancelling 2ab from both sides gives a² + b² = c².

That algebraic cancellation is the entire theorem — clean enough to reproduce in an exam.

Solved examples

Example 1: Finding the hypotenuse

A right triangle has legs of 6 cm and 8 cm. Find the hypotenuse.

Using c² = a² + b² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm. This 6-8-10 triangle is just a scaled-up 3-4-5 triangle, one of the most common patterns in exams.

Example 2: Finding a missing leg

A ladder 13 m long leans against a wall and its foot is 5 m from the wall. How high up the wall does it reach?

Here the ladder is the hypotenuse: 13² = 5² + h², so h² = 169 − 25 = 144 and h = 12 m. The 5-12-13 triple appears constantly, so memorise it.

Pythagorean triples worth memorising

A Pythagorean triple is a set of three whole numbers that satisfy the theorem. Recognising them saves time:

Any multiple of a triple is also a triple, so 6-8-10, 9-12-15, and 10-24-26 all work.

The converse of the theorem

The converse is equally examinable: if the square of the longest side of a triangle equals the sum of the squares of the other two sides, then the triangle is right-angled. This is how you prove a triangle has a 90° angle without measuring it — just test whether a² + b² = c².

Where it is used in real life

• Construction: builders use the 3-4-5 rule to set out perfect right angles for walls and foundations.
• Navigation and maps: the straight-line distance between two points is found with the distance formula, which is the Pythagoras theorem in disguise.
• Screens and displays: a TV's "42-inch" size is the diagonal — the hypotenuse of its width and height.

Common mistakes to avoid

• Applying the theorem to a triangle that is not right-angled. It only works when there is a 90° angle.
• Treating a leg as the hypotenuse. The hypotenuse is always opposite the right angle and is the longest side.
• Forgetting to take the square root at the end — you find c² first, then c.

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