How do you prove the Pythagoras theorem?

In one line

The rule

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. (NCERT)

Formula: a² + b² = c² Where:

• a and b are the lengths of the two legs (sides meeting at the right angle)
• c is the length of the hypotenuse (side opposite the right angle)

Let's understand it simply

Imagine you are drawing a right-angled triangle on your maths notebook. The longest side, opposite the 90° angle, is called the hypotenuse. The other two sides are called legs. Now, think of making a square on each side of the triangle. The area of the square on the hypotenuse is always equal to the total area of the squares on the other two sides.

It's like this: if you have a triangle with sides 3 cm and 4 cm, and you make squares on those sides, their areas are 9 cm² and 16 cm². Add them: 25 cm². The square on the hypotenuse will also be 25 cm², so its side is 5 cm. This is the magic of the Pythagorean theorem.

How it works

Let's prove the Pythagorean theorem using a simple geometric method (using squares):

1. Draw a right-angled triangle with sides a, b, and hypotenuse c.
2. Make a square with side length (a + b).
3. Inside this big square, arrange four identical right-angled triangles (each with sides a, b, c) so that their hypotenuses form a smaller square in the center.
4. The area of the big square is (a + b)².
5. The area is also equal to the area of the four triangles plus the area of the small inner square (which has side c).
6. Area of one triangle = ½ × a × b. So, four triangles = 4 × ½ × a × b = 2ab.
7. Area of the small inner square = c².
8. So, total area = area of triangles + area of inner square: 2ab + c².
9. But we also know the total area is (a + b)² = a² + 2ab + b².
10. Set the two expressions for area equal: a² + 2ab + b² = 2ab + c².
11. Subtract 2ab from both sides: a² + b² = c².

The story behind it

The Pythagorean theorem is named after Pythagoras, a Greek mathematician from around 500 BCE. But, Indian mathematicians like Baudhayana and Apastamba knew about this relationship much earlier, as seen in ancient texts. They used it for building altars and in geometry. The Greeks made it famous, but the idea was known in India, Babylon, and China too.

One memorable detail: Pythagoras is said to have celebrated his discovery by sacrificing an ox, but Indian mathematicians used it for practical problems like land measurement.

Where you see it in real life

• Cricket pitch marking: The diagonal distance across a rectangular pitch uses the theorem to measure accurately.
• Ladders against a wall: If you know how high you want the ladder to reach and how far it stands from the wall, you can find the ladder's length.
• Cycle rides: If you ride 3 km east and then 4 km north, your direct distance from the start is 5 km (using 3² + 4² = 5²).
• TV or phone screens: The size (in inches) is the diagonal, calculated using the length and width.
• Construction: Builders use ropes with knots at 3, 4, and 5 units to make perfect right angles.
• Maps and navigation: Finding the shortest path between two points on a grid.

Why it matters

• Engineering and architecture: Used in designing buildings, bridges, and roads.
• Navigation and GPS: Calculating shortest distances.
• Computer graphics: For calculating distances between pixels.
• Foundation for trigonometry: Later chapters in maths and physics use this theorem.

Worked example

Let's prove it with numbers:

Given: a = 6 cm, b = 8 cm
Step 1: Write the rule        a² + b² = c²
Step 2: Put in the numbers    6² + 8² = c²
Step 3: Calculate squares     36 + 64 = c²
Step 4: Add                   100 = c²
Step 5: Find c                c = √100 = 10 cm

So, the hypotenuse is 10 cm.

Common mistakes

• Forgetting it only works for right-angled triangles: The theorem does NOT work for triangles without a 90° angle.
• Mixing up which side is the hypotenuse: Always the side opposite the right angle, and always the longest.
• Adding instead of squaring: Don't just add side lengths; you must add their squares.
• Using wrong units: All sides must be in the same unit (cm, m, etc.).

Remember it

"In a right triangle, the big side’s square equals the sum of the other two squares." Or: "a² + b² = c² — only for right triangles!"