What does the Pythagoras theorem actually say?

In one line

The rule

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.

The formula is: a² + b² = c²

• a = length of one side (called a "leg")
• b = length of the other side (the other "leg")
• c = length of the hypotenuse (the longest side, opposite the right angle)

Let's understand it simply

Imagine you are drawing a right-angled triangle on your notebook. The right angle is like the corner of your maths book (90°). The side across from this right angle is always the longest. This is called the hypotenuse (say: hy-pot-en-use).

Suppose you have a triangle with sides 3 cm and 4 cm, and you want to find the hypotenuse. The Pythagoras theorem tells you that if you make a square on each side, the area of the biggest square (on the hypotenuse) will be exactly equal to the sum of the areas of the other two squares. It is like saying: if you have a cricket pitch 3 m wide and 4 m long, the straight line from one corner to the opposite corner is 5 m — not 7 m, but 5 m!

How it works

1. Check for a right angle: Make sure your triangle has a 90° angle.
2. Identify the sides: Find the two shorter sides (legs) and the longest side (hypotenuse).
3. Square the lengths: Multiply each side by itself (for example, 3 × 3 = 9).
4. Add the squares of the legs: Add the squares of the two shorter sides.
5. Compare or find the hypotenuse: The sum equals the square of the hypotenuse. If you know two sides, you can find the third.

The story behind it

The theorem is named after Pythagoras, a Greek mathematician who lived around 2,500 years ago. But, Indian mathematicians like Baudhayana knew about this rule much earlier (over 3,000 years ago). They used it to build perfect right angles for making altars. Pythagoras made it famous in Greece, but the idea was known in India, Babylon, and China too. One memorable detail: In olden days, people used ropes with 12 equal knots (3+4+5) to make right angles for building walls!

Where you see it in real life

• Cricket pitch: Measuring the shortest path from one wicket to the opposite corner of the boundary — use the theorem to find the diagonal.
• Cycle ride: If you go 6 km east and then 8 km north, the straight shortcut home is 10 km (6² + 8² = 10²).
• Ladders: Placing a ladder against a wall — the ladder, wall, and ground make a right triangle.
• TV screens: The size (in inches) is the diagonal, not the height or width. Use the theorem to find it.
• Plotting land: Farmers use it to measure the diagonal of rectangular fields.
• Building construction: Making sure corners are exactly 90°, using the 3-4-5 method.

Why it matters

• Geometry and Trigonometry: Many topics in maths and science use this rule as a base.
• Engineering and Architecture: Used to design buildings, bridges, and roads.
• Navigation: Helps pilots, sailors, and drivers find the shortest route.
• Exams: Questions on triangles, distance, and area often use this theorem.

Worked example

Let's solve: A right-angled triangle has sides 5 cm and 12 cm. What is the length of the hypotenuse?

Given: a = 5 cm, b = 12 cm
Step 1: Write the rule        a² + b² = c²
Step 2: Put in the numbers    5² + 12² = c²
Step 3: Calculate squares     25 + 144 = c²
Step 4: Add                  169 = c²
Step 5: Find square root     c = √169 = 13 cm

So, the hypotenuse is 13 cm.

Common mistakes

• Mixing up the sides: The hypotenuse is always opposite the right angle and is the longest side.
• Using the rule for non-right triangles: This theorem only works for right-angled triangles.
• Forgetting to take the square root: After adding the squares, you must find the square root to get the length.
• Adding the sides, not their squares: The theorem uses squares of the sides, not just their lengths.

Remember it

"In a right triangle, a² + b² = c² — the sum of the squares of the two shorter sides equals the square of the longest side."

A memory trick: Think of a triangle with sides 3, 4, and 5. 3² + 4² = 9 + 16 = 25 = 5². This is the classic example!