The Pythagorean Theorem Explained
6 min read · Updated September 26, 2026 · by the SolveCalcPro editorial team
In every right triangle, the square of the longest side (the hypotenuse) equals the sum of the squares of the two shorter sides: a² + b² = c². The result is over 2,500 years old and still used daily in building, navigation and computer graphics.
The formula and what each letter means
Label the two sides that meet at the right angle a and b (the legs) and the side opposite the right angle c (the hypotenuse). Then c² = a² + b². The hypotenuse is always the longest side.
Why it works: a picture proof
Take four copies of the triangle and arrange them inside a square of side (a + b). The leftover space is a tilted square of side c, so its area is c². Rearranging the same four triangles leaves two squares of areas a² and b². Because the total area is the same in both arrangements, c² = a² + b².
Finding a side
Hypotenuse: legs 9 and 12 give c = √(81 + 144) = √225 = 15.
A leg: a 17 ft ladder reaching 15 ft up a wall has its base √(17² − 15²) = √64 = 8 ft from the wall.
The Pythagorean theorem calculator does both and adds the area and angles.
Pythagorean triples
Whole-number sets that work: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Multiples of a triple also work, so 6-8-10 and 9-12-15 are triples too. Carpenters use 3-4-5 to make square corners.
Testing for a right angle
The converse is also true: if a² + b² = c² for the three sides of a triangle, it has a right angle. If a² + b² is greater than c², the angle is acute; if less, obtuse.
Distance between two points
On a coordinate grid, the distance between (x₁, y₁) and (x₂, y₂) is √((x₂−x₁)² + (y₂−y₁)²). From (1, 2) to (4, 6): √(9 + 16) = 5.
Practice questions
Test yourself, then tap to check. Answers are calculated by the tools linked below.
- A right triangle has legs 9 and 12. Find the hypotenuse.
Show answer
Hypotenuse c: 15
- Area of a circle with radius 7.
Show answer
Area: 153.93804
- Solve x² − 7x + 10 = 0.
Show answer
x₁: 5 · x₂: 2
Frequently asked questions
Does the theorem work for any triangle?
Only right triangles. Other triangles use the law of cosines, which reduces to the Pythagorean theorem when the angle is 90°.
Who discovered it?
It is named for Pythagoras, but Babylonian and Indian mathematicians knew it earlier.
Can the sides be decimals?
Yes. Any positive lengths that satisfy the equation work.
What is the 3D version?
The diagonal of a box is √(a² + b² + c²).
Try it yourself
Pythagorean Theorem CalculatorFind the hypotenuse or a missing leg of a right triangle, plus area, perimeter and angles.Geometry CalculatorArea, perimeter, circumference, volume and surface area of common shapes.
Quadratic Equation SolverSolve ax² + bx + c = 0 with the discriminant, roots and vertex.Keep learning
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