The Pythagorean theorem links the sides of any right triangle: the squares of the two shorter sides add up to the square of the longest side. It is among the most practical ideas in geometry.
The formula
If a and b are the legs and c is the hypotenuse, then a² + b² = c². To find a missing leg, rearrange it: a = √(c² − b²).
Example: how long a ladder?
To reach a window 12 feet up with the base 5 feet from the wall, the ladder needs to be √(5² + 12²) = √169 = 13 feet. Leave extra length for safe overlap and follow the ladder manufacturer's angle guidance.
Checking a square corner
Builders use the 3-4-5 rule. Mark 3 units along one edge and 4 units along the other. If the diagonal between the marks measures exactly 5, the corner is square. Multiples such as 6-8-10 work too.
Distance between two points
The distance formula is the same idea. For points (x₁, y₁) and (x₂, y₂), the distance is √((x₂ − x₁)² + (y₂ − y₁)²).
Common mistakes
The theorem only applies to right triangles. The hypotenuse is always the longest side, so do not subtract when you should add. Keep units consistent before squaring.
Pythagorean triples
Whole-number sets that satisfy the theorem include 3-4-5, 5-12-13, 8-15-17, and 7-24-25. Any multiple of a triple also works. They are useful because you can check a right angle with whole-number measurements and no square roots.
Example: a screen diagonal
A 16:9 screen that is 48 inches wide is 27 inches tall. Its diagonal is √(48² + 27²) = √3,033 ≈ 55.1 inches, which is why it is sold as a 55-inch screen. The theorem lets you compare screens of different shapes by width and height.
Ramps and slopes
If a ramp rises 30 inches and uses a 1:12 slope (the maximum commonly cited for accessible ramps in the United States), the run is 360 inches, or 30 feet. The ramp surface length is √(30² + 360²) ≈ 361.2 inches. Always check local building codes for actual requirements.
Going to three dimensions
The diagonal of a box with length l, width w, and height h is √(l² + w² + h²). For a 3 by 4 by 12 box, that is √(9 + 16 + 144) = 13 units. This is useful for checking whether a long item will fit inside a box or suitcase.
Using the converse
If the three sides of a triangle satisfy a² + b² = c², the triangle has a right angle. This is the reason the 3-4-5 method works for squaring corners.
Try the calculators
Pythagorean Theorem Calculator
This guide is for general information only. It is not financial, medical, or legal advice.