Pythagorean Theorem Calculator
Calculate the missing side of a right triangle — enter the two sides you know and leave the third at 0.
How this calculator works
The Pythagorean theorem relates the three sides of any right triangle: the sum of the squares of the two legs equals the square of the hypotenuse.
a and b are the two legs; c is the hypotenuse (the longest side, opposite the right angle).
Real-world applications of the Pythagorean theorem
Beyond geometry class, the Pythagorean theorem is used constantly in construction (checking that corners are truly square by verifying a 3-4-5 triangle), navigation (calculating shortest-path distances), and even TV and monitor sizing (screen sizes are measured diagonally, which is the hypotenuse of the screen's rectangle).
The '3-4-5 rule' used by builders is a direct application — if two sides of a corner measure 3 and 4 units, the diagonal should measure exactly 5 units for the corner to be a true right angle.
A worked example
For a right triangle with legs 6 and 8: the hypotenuse is the square root of (6 squared plus 8 squared), or the square root of 100, which is 10 — the classic 6-8-10 triangle, a scaled version of the well-known 3-4-5 right triangle.
For legs 5 and 12 instead: the hypotenuse is the square root of (25 plus 144), or the square root of 169, which is 13 — another commonly referenced 'Pythagorean triple' where all three sides are whole numbers.
Real-world uses of the Pythagorean theorem
Beyond geometry class, the Pythagorean theorem shows up in practical measurement problems: finding the diagonal distance across a rectangular space, checking whether a corner is truly square (a 3-4-5 triangle is a classic carpenter's trick), calculating cable or ladder lengths, and navigation distance calculations all rely on this same relationship.
The '3-4-5 triangle' is a well-known example where all three sides are whole numbers (3² + 4² = 5², or 9 + 16 = 25) — useful as a quick mental check or practical measuring technique in construction and design.
Frequently asked questions
Does this only work for right triangles?
Yes — the Pythagorean theorem specifically applies to right triangles, where one angle is exactly 90 degrees.
Does the Pythagorean theorem work for any triangle?
No — it only applies to right triangles, where one angle is exactly 90 degrees. For other triangles, different formulas like the law of cosines are needed.
How is this used to check if a corner is 'square' in construction?
Builders measure 3 units along one wall and 4 units along the adjacent wall, then check that the diagonal between those points measures exactly 5 units — confirming a true 90-degree angle.
What is a 3-4-5 triangle used for?
It's a simple, practical way to check if a corner is a true right angle — measuring 3 units along one side, 4 along the other, and confirming the diagonal is exactly 5 units confirms a 90-degree angle, a technique commonly used in construction.