Pythagoras Theorem Explained with Examples
The Pythagoras theorem is one of the most important results in geometry, and it appears in CBSE and ICSE papers from Class 9 right up to JEE level. It connects the three sides of a right-angled triangle with a single, elegant equation. This article explains the theorem, the intuition behind it, useful triples and real-life applications.
What the theorem says
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. If the two shorter sides (legs) are a and b, and the hypotenuse is c, then:
- a² + b² = c²
The hypotenuse is always the side opposite the 90° angle and is the longest side. This relationship is the foundation for distance calculations in coordinate geometry.
Proof intuition
Imagine drawing a square on each side of the right triangle. The theorem says the area of the square built on the hypotenuse exactly equals the combined area of the two squares on the legs. A simple way to see this: take four identical right triangles and arrange them inside a large square of side (a + b). The empty space in the middle forms a tilted square of area c². Rearranging the same four triangles leaves two empty squares of areas a² and b². Since the total area is unchanged, a² + b² = c².
Pythagorean triples
A Pythagorean triple is a set of three whole numbers that satisfy the theorem. Memorising a few saves time in exams:
- 3, 4, 5: since 9 + 16 = 25.
- 5, 12, 13: since 25 + 144 = 169.
- 8, 15, 17: since 64 + 225 = 289.
- 7, 24, 25: since 49 + 576 = 625.
Any multiple of a triple is also a triple, so 6, 8, 10 works too. Keeping these handy is a great memory habit; see our guide on how to memorise maths formulas.
Worked example 1
A ladder leans against a wall. Its foot is 6 m from the wall and the ladder is 10 m long. How high up the wall does it reach?
Here c = 10 (hypotenuse), a = 6, and b is the height. So 6² + b² = 10², giving 36 + b² = 100, so b² = 64 and b = 8 m. The ladder reaches 8 m up the wall.
Worked example 2
Find the length of the diagonal of a rectangle that is 9 cm long and 12 cm wide.
The diagonal is the hypotenuse: c² = 9² + 12² = 81 + 144 = 225, so c = √225 = 15 cm.
Real-life uses
The theorem is used by builders to check that corners are square, by surveyors to measure distances, in navigation, and in computer graphics. You will also use it constantly when finding distances in Class 10 Maths and trigonometry.
Frequently asked questions
Does the Pythagoras theorem work for all triangles?
No. It only applies to right-angled triangles, where one angle is exactly 90°. For other triangles you need the cosine rule, which is studied later.
How do I know which side is the hypotenuse?
The hypotenuse is always opposite the right angle and is the longest side of the triangle. In the equation a² + b² = c², c always represents the hypotenuse.
Can the theorem be used to check if a triangle is right-angled?
Yes. If the three sides satisfy a² + b² = c² (with c the longest side), the triangle must be right-angled. This is called the converse of the Pythagoras theorem.