The Pythagorean Theorem: Concept, Proof Idea and Uses
What the theorem actually says
A right triangle is a triangle with one angle of exactly 90° — a square corner. The two sides that form that corner are called the legs, and the side opposite the corner is the hypotenuse, which is always the longest side. If the legs are a and b and the hypotenuse is c, the theorem says that the squares of the two legs add up to the square of the hypotenuse.
The rule comes with a strict condition: it only works on right triangles. The theorem also runs in reverse — if the three sides of any triangle satisfy a² + b² = c², then the triangle must contain a right angle. Builders use that reverse version constantly to check that a corner is truly square before pouring a foundation or framing a wall.
The formula in action
The formula links all three sides, so knowing any two of them locks in the third. To find the hypotenuse, add the squares of the legs and take the square root. To find a missing leg, subtract the square of the known leg from the square of the hypotenuse, then take the square root.
a² + b² = c²The classic 3-4-5 triangle
A right triangle with legs 3 and 4 gives 3² + 4² = 9 + 16 = 25, and 25 = 5², so the hypotenuse is exactly 5.
Whole-number combinations like this are called Pythagorean triples, and 3-4-5 is the smallest one. Any multiple of it also works: 6-8-10, 9-12-15, 30-40-50.
c = √25 = 5
A 5 m ladder reaching 4 m up a wall
The ladder is the hypotenuse (5 m) and the wall height is one leg (4 m). The wall meets the ground at 90°, so the distance from the wall to the foot of the ladder satisfies base² = 5² − 4² = 25 − 16 = 9.
The base is the square root of 9, so the ladder's foot sits exactly 3 m from the wall.
base = √9 = 3 m
A straight-line distance
You walk 30 m east, then 40 m north. The two legs of a right triangle are 30 and 40, and your straight-line distance from the start is the hypotenuse: √(30² + 40²) = √(900 + 1600) = √2500 = 50 m.
This is exactly how mapping apps combine an east-west step and a north-south step into one distance.
distance = √2500 = 50 m
A proof you can see: rearranging squares
The most famous proof needs no algebra — only a picture and the fact that the same pieces must cover the same area. Draw a square on each side of a right triangle. The two smaller squares have areas a² and b²; the big tilted square on the hypotenuse has area c². The claim is simply that the two small squares together fill exactly the same area as the big one.
Here is the rearrangement. Take a large square whose sides measure (a + b) and place four identical copies of the triangle inside it. Arrange the four triangles in two columns down the middle: the leftover space splits into two squares, of areas a² and b². Now rearrange the same four triangles in the same big square, each tucked into a corner: the leftover space becomes a single tilted square of area c². Nothing was added and nothing was removed — same outer square, same four triangles — so the leftover area must be equal both times.
That is the entire argument: a² + b² = c². Versions of this picture appear in ancient Chinese and Indian mathematics, centuries before the rule was named after the Greek philosopher Pythagoras, who lived from about 570 to 495 BC.
Pythagorean triples worth memorizing
Triples save time because they avoid square roots entirely: when a problem hands you two sides of a triple, the third side is a whole number. Each triple can also be scaled by any factor and it stays valid. To verify any candidate, just check that the two smaller squares add up to the largest one.
| Leg a | Leg b | Hypotenuse c | Check |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 |
| 5 | 12 | 13 | 25 + 144 = 169 |
| 8 | 15 | 17 | 64 + 225 = 289 |
| 7 | 24 | 25 | 49 + 576 = 625 |
| 20 | 21 | 29 | 400 + 441 = 841 |
Where the theorem shows up
Construction. To square a corner, carpenters measure 3 units along one wall and 4 units along the other, then adjust the corner until the diagonal between those two points measures exactly 5 units. Sides that satisfy 3-4-5 guarantee the 90° angle — the converse of the theorem doing real work.
Navigation and distance. Any trip with an east-west leg and a north-south leg is a right triangle. Walk 30 m east and 40 m north, and you are 50 m from where you started, as the crow flies. GPS and surveying equipment compute right-triangle distances like this thousands of times per route.
Screens. Television and monitor sizes are measured diagonally, because the diagonal is the hypotenuse of the screen's width and height. A 16:9 screen that is 16 inches wide and 9 inches tall is sold as √(16² + 9²) = √337 ≈ 18.4 inches — the size in the store label refers to the diagonal, never the width.
Two mistakes to avoid
First, confirm the right angle before anything else. The theorem gives wrong answers on triangles without a 90° angle: an equilateral triangle with sides 2 does not have a side of length √(2² + 2²) ≈ 2.83, because it has no right angle at all.
Second, keep the hypotenuse in its place: it is always the side opposite the right angle, never one of the sides forming it. When you solve for the hypotenuse, the squares add; when you solve for a leg, one square gets subtracted. Mixing up those two operations is the most common source of wrong answers in homework and on job sites alike.