Pythagorean Theorem

Stories say that four thousand years ago, Egyptian builders would stretch a rope divided into 12 equal parts into a 3-4-5 triangle ( $3+4+5=12$ , using every part) — and whenever the sides came out in that ratio, the corner was a perfect right angle. They knew no algebra, yet they were already using the Pythagorean theorem.

Meet the right triangle

A right triangle is a triangle with one angle of exactly 90°. Its three sides have names:

The two sides that touch the right angle are the legs;
The side opposite the right angle is the hypotenuse — always the longest one.

The theorem: a tale of three squares

Build a square on each of the three sides. Over two thousand years ago, followers of Pythagoras noticed a relationship that never fails:

$a^2 + b^2 = c^2$

The areas of the two squares on the legs add up to exactly the area of the big square on the hypotenuse. With legs 3 and 4: $3^2 + 4^2 = 9 + 16 = 25 = 5^2$ , so the hypotenuse is 5.

The relationship also runs in reverse: if the squares of the two shorter sides add up to the square of the longest side, the triangle must contain a right angle — no protractor needed, just arithmetic. Followers of Pythagoras even treated it as a jigsaw puzzle: cut up the two squares on the legs and their pieces can always be rearranged into the big square on the hypotenuse.

Drag the legs around and watch the three squares respond:

InteractivePythagoras Explorer

Leg a3Leg b4

3² + 4² = 9 + 16 = 25 = c²

Hypotenuse c c = √25 ≈ 5.00整数勾股数 · Pythagorean triple!

Pythagorean triples: ready-made trios

Right triangles whose three sides are all whole numbers come in famous combos called Pythagorean triples:

$3, 4, 5$ : because $9 + 16 = 25$ ;
$6, 8, 10$ : scale 3-4-5 up by 2 and the right angle survives;
$5, 12, 13$ : because $25 + 144 = 169$ .

And the list never runs dry: multiply 3-4-5 by 3 to get 9-12-15, by 4 to get 12-16-20 — every scaled copy works, because multiplying just enlarges the triangle.

Memorize 3-4-5 and you can pull a right angle out of your pocket on a building site, a floor plan or a grid.

Worked example: a ladder against a wall

Ladder against a wall

A 5-meter ladder leans against a wall with its foot 3 meters from the base. How high up the wall does the top reach?

The ladder is the hypotenuse $c = 5$ , the distance from the wall is a leg $a = 3$ , and the height is $b$ :

$b^2 = c^2 - a^2 = 25 - 9 = 16, \qquad b = 4$

The top sits 4 meters above the ground. To find a leg you subtract; to find the hypotenuse you add. Identify the hypotenuse first and the formula never gets flipped.

Right triangles only

The Pythagorean theorem is the exclusive property of right triangles. In a triangle with sides 4, 5 and 6, $4^2 + 5^2 = 41 \neq 36$ — if the triangle is not right, the formula does not apply. Confirm that 90° angle exists before you compute.

Check yourself

Quick quiz

1. A right triangle has legs 6 and 8. How long is the hypotenuse?
2. Is a triangle with sides 5, 12 and 13 a right triangle?
3. Which triangles does the Pythagorean theorem apply to?