What does the Pythagoras theorem 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 and b are the lengths of the two shorter sides (legs)
• c is the length of the hypotenuse (the side opposite the right angle)

Let's understand it simply

Imagine you are drawing a triangle on the ground with a right angle (like the corner of a book). The side across from this right angle is the hypotenuse (the longest side). If you make a square on each side of the triangle, the area of the square on the hypotenuse will be exactly the same as the total area of the squares on the other two sides.

Think of it like this: If you have a triangle with sides of 3 feet and 4 feet, and you want to know the length of the slanting side (hypotenuse), the Pythagoras theorem helps you find it easily, just by adding and squaring numbers.

How it works

1. Identify the right-angled triangle. The right angle is 90°.
2. Label the two shorter sides as a and b.
3. Label the longest side (opposite the right angle) as c (the hypotenuse).
4. Square the lengths of both shorter sides: calculate a² and b².
5. Add these two squares: a² + b².
6. The answer is equal to c². To find c, take the square root: c = √(a² + b²).

The story behind it

The Pythagoras theorem is named after Pythagoras, a Greek mathematician from around 2500 years ago. But, Indian mathematicians like Baudhayana and ancient Egyptians also knew about this rule even before Pythagoras. They used it for building and land measurement. Pythagoras made the rule famous in Greece, but it is really a gift from many cultures.

Where you see it in real life

• Cricket pitch marking: When marking the crease, groundsmen use ropes to make sure the lines are straight and at right angles. The Pythagoras theorem helps check the diagonal distance.
• Building a rectangular room: To check if the corners are exactly 90°, workers measure the diagonals. If the diagonal matches the value from the theorem, the room is square.
• Ladders against a wall: If a 5-foot ladder reaches a wall 4 feet high and is 3 feet away from the wall, the theorem checks if it is safe.
• Finding shortest paths: When you cut across a field instead of walking along the edges, you are using the hypotenuse—the shortest distance.
• TV or phone screens: The size of the screen (like 32 inches) is the diagonal, found using the Pythagoras theorem.
• Kite flying: The height of the kite, the length of the string, and the distance from the flyer form a right triangle.

Why it matters

• Used in construction, to make sure buildings are straight and safe.
• Needed in navigation and map-making to find distances.
• Important for physics and engineering problems involving distances and forces.
• Forms the base for trigonometry and many geometry questions in exams like CBSE, JEE, and NEET.

Worked example

Let's say you have a right-angled triangle with sides 6 cm and 8 cm. What is the length of the hypotenuse?

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: Take square root      c = √100
Step 6: Answer                c = 10 cm

Common mistakes

• Mixing up the sides: The hypotenuse is always the side opposite the right angle and is the longest side.
• Forgetting to square or square root: Some students add the numbers directly, not their squares, or forget to take the square root at the end.
• Using the theorem for non-right triangles: The rule only works for right-angled triangles.
• Wrong units: Always keep the units the same for all sides (all in cm, m, etc.).

Remember it

"Square on the hypotenuse equals sum of squares on the other two sides."

Or: "a² + b² = c² — only for right triangles!"