Find the missing side of a right triangle — hypotenuse or a leg.
a² + b² = c², where c is the hypotenuse (the side opposite the right angle).
The Pythagorean theorem — a² + b² = c², where c is the hypotenuse of a right triangle — is named for Pythagoras, but evidence shows Babylonian mathematicians were already using the same relationship well over a thousand years before him, and independent versions of it were also discovered in ancient China and India. What made the Greek treatment distinctive wasn't the discovery of the relationship itself but the proof — a rigorous demonstration that the relationship holds for every right triangle, not just specific cases that happened to work, which is a genuinely different mathematical achievement than simply noticing the pattern.
Its lasting practical value is that it converts a diagonal-distance problem into simple arithmetic on two perpendicular measurements — which is why it shows up constantly outside pure geometry: finding the diagonal length of a rectangular room or TV screen, calculating the shortest straight-line distance between two points on a map given their horizontal and vertical separation, checking whether a corner is truly square in construction (a 3-4-5 triangle, satisfying 3²+4²=5², is a classic on-site method builders still use to verify a right angle without any specialized tools), and calculating straight-line distance in coordinate geometry and physics. Any time a problem involves a right angle and two known perpendicular measurements, the theorem turns what looks like it might need trigonometry into a single, exact calculation.
a² + b² = c², where a and b are the two shorter sides (legs) of a right triangle and c is the hypotenuse (the longest side, opposite the right angle). Given any two of the three sides, it calculates the missing third side exactly.
The relationship was known and used by Babylonian mathematicians well over a thousand years before Pythagoras, and was independently discovered in ancient China and India as well. What's specifically credited to the Greek tradition is a rigorous proof that the relationship holds for every right triangle, not just the discovery of the pattern itself.
Because 3² + 4² = 5² (9 + 16 = 25) satisfies the Pythagorean theorem exactly, so a triangle with sides of 3, 4 and 5 units (or any multiple of those ratios) must contain a genuine 90-degree angle. Measuring out those proportions on-site is a simple, tool-free way to verify a corner is truly square during construction.
Yes — it's the basis for calculating straight-line distance between two points given their horizontal and vertical separation, finding the diagonal of a rectangle or screen, and it underlies the distance formula used throughout coordinate geometry and physics. Any right-angle relationship between two perpendicular measurements can be solved with it.