What the Theorem Actually Says
Take any right triangle - one with exactly one 90-degree angle. The two shorter sides that form that right angle are called legs (labelled a and b). The side directly opposite the right angle, always the longest of the three, is the hypotenuse (labelled c).
Where:
a = one leg
b = other leg
c = hypotenuse (the side opposite the right angle)
So: square each leg, add the results, and you get exactly the square of the hypotenuse. Not approximately. Exactly. This holds for every right triangle regardless of size or orientation. The catch is strict - the triangle must have a 90-degree angle. For other triangles, you'd use the Law of Cosines instead.
Why It's True: The Square Areas Proof
The most satisfying proof is geometric. Draw a square on each side of a right triangle - one on leg a, one on leg b, one on the hypotenuse c. The theorem says the area of the square on the hypotenuse equals the combined areas of the squares on the two legs.
Square on leg a (side 3): area = 3² = 9
Square on leg b (side 4): area = 4² = 16
Square on hypotenuse c: area = 9 + 16 = 25
Side length of that square: √25 = 5
So the hypotenuse = 5
The classic rearrangement proof works like this: start with a large square of side (a + b) and tile it two different ways using four copies of the right triangle. In one arrangement, the four triangles sit around a tilted central square of side c - the leftover area is c². In the other arrangement, the same four triangles leave behind two separate squares of sides a and b. Since the outer square and the four triangles are identical in both arrangements, the leftover areas have to be equal: c² = a² + b². That's not hand-waving - it's a clean, rigorous argument.
Working Out Missing Sides
Case 1: Both legs are known, find the hypotenuse.
Legs 5 and 12:
c = √(5² + 12²) = √(25 + 144) = √169 = 13
Case 2: One leg and the hypotenuse are known, find the missing leg.
Rearrange the formula: a² = c² − b², so a = √(c² − b²).
other leg = √(17² − 8²) = √(289 − 64) = √225 = 15
When the answer isn't a perfect square, just leave it as a simplified radical or use a calculator. Legs of 2 and 3 give a hypotenuse of √13, which is irrational - roughly 3.606. That's completely fine. Most right triangles produce irrational answers. The clean integer cases are the exceptions, and those special sets are called Pythagorean triples.
Pythagorean Triples: The Integer Special Cases
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c² exactly. They come up constantly in textbook problems and standardized tests because they give clean integer answers. Learn the primitive triples - the ones that share no common factor - and you'll recognize them on sight without needing a calculator at all.
| a | b | c | Verification |
|---|---|---|---|
| 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 ✓ |
Any multiple of a Pythagorean triple is also a triple. The 3-4-5 family alone generates 6-8-10, 9-12-15, 12-16-20, and so on - just multiply all three sides by the same integer. And this isn't purely academic. Construction workers use the 3-4-5 triple to check right angles in the field: measure 3 units along one wall, 4 along the adjacent wall, and if the diagonal is exactly 5, the corner is square. Simple, reliable, and it's been used in building for thousands of years.
The Distance Formula Is Just Pythagoras in Disguise
The formula for the straight-line distance between two points on a coordinate grid isn't a separate theorem - it's the Pythagorean theorem applied to a right triangle formed by the horizontal and vertical gaps between the points.
Given points P₁(x₁, y₁) and P₂(x₂, y₂):
Vertical leg: Δy = y₂ − y₁
Distance (hypotenuse): d = √(Δx² + Δy²)
Distance from (2, 1) to (7, 13):
Δx = 7 − 2 = 5
Δy = 13 − 1 = 12
d = √(5² + 12²) = √(25 + 144) = √169 = 13
Notice the 5-12-13 triple appearing naturally there. Whenever the horizontal and vertical separations happen to form a Pythagorean triple, the distance is an integer - a useful sanity check. And this idea scales up. GPS receivers use a three-dimensional version of the same logic: the distance from a receiver to each satellite forms the hypotenuse of a triangle in 3D space, and intersecting those distances pins down the receiver's location. A theorem Pythagoras worked out on flat triangles over 2,000 years ago ends up doing the math inside your phone.
The CalcSolver Pro triangle calculator applies the Pythagorean theorem - and the full Law of Cosines for non-right triangles - to find any missing side or angle when you supply what you know.