The Pythagorean Theorem
“In a right triangle, knowing two sides fixes the third.”
The formula
a² + b² = c²How to read it: in a right triangle, squaring the two legs and adding them equals the square of the hypotenuse
- a, b
- — the two sides that form the right angle (the legs)
- c
- — the longest side, opposite the right angle (the hypotenuse)
- a², b², c²
- — the area of a square built on each side
The hook
Given just one right angle, you can compute the third side from the other two instead of measuring it.
In plain words
In a right triangle, square each of the two shorter sides and add them, and you land exactly on the square of the hypotenuse.
The intuition
Draw a square on each of the three sides. The two smaller squares' areas (a², b²) fit together perfectly to fill the big square (c²). It isn't a relationship of lengths but a 'balance of areas'.
How it's built
The key is the right angle. Only when the angle is 90° does the sum of the two squared sides equal the square of the hypotenuse. The hypotenuse c is always opposite the right angle — the longest side.
Example
If the two legs are 3 and 4, then 3² + 4² = 9 + 16 = 25. The hypotenuse is √25 = 5. That's why 3-4-5 is the most famous right triangle.
Common mistake
Don't plug just any side in for the hypotenuse. c is always 'the side opposite the right angle, the longest one'. And it's a relationship between squares, not a + b = c.
Where it's used
Distance between two points on a map, a screen's diagonal size, checking a building is vertical, the distance formula on the coordinate plane — nearly anywhere a 'straight-line distance' is needed.
Where it came from
It bears Pythagoras's name, but Babylonians, Egyptians, Chinese, and Indians already knew and used this relationship thousands of years earlier.
Prerequisites
Quick check
In a right triangle with legs 6 and 8, what is the hypotenuse?
- 10✓
- 12
- 14
- 48
Practice
What is the hypotenuse of a right triangle with legs 5 and 12?
Answer: 13
- 5² + 12² = 25 + 144 = 169
- hypotenuse = √169 = 13
Takeaway: 5-12-13 is another whole-number right triangle you'll see often.
A right triangle has hypotenuse 13 and one leg 5. What is the other leg?
Answer: 12
- When you know the hypotenuse, you subtract: b² = c² − a²
- b² = 13² − 5² = 169 − 25 = 144
- b = √144 = 12
Takeaway: Knowing the hypotenuse and one leg, subtract — don't add.
What is the diagonal of a square with side length 1? (give an approximate value)
Answer: 1.41
- The diagonal is the hypotenuse of a right triangle with legs 1 and 1
- 1² + 1² = 2, so the diagonal = √2
- √2 ≈ 1.414
Takeaway: A square's diagonal is √2 times its side — an irrational, non-terminating number.