// maths › Triangles
Find the missing side of a right-angled triangle using Pythagoras' theorem, a squared plus b squared equals c squared, with full working and a visual proof that the two smaller squares add up to the largest.
a² + b² = c²
A mind behind this: Pythagoras of Samos c. 570 – c. 495 BC
Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides. Written as a² + b² = c², it lets you find any one side when you know the other two.
Square each leg, add the results, then take the square root. For example, with legs of 3 and 4: 3² + 4² = 9 + 16 = 25, and the square root of 25 is 5, so the hypotenuse is 5.
Yes. Rearrange the formula to leg = √(c² − other²). Subtract the known leg's square from the hypotenuse's square, then take the square root. With a hypotenuse of 13 and one leg of 5: √(169 − 25) = √144 = 12.
No. Pythagoras' theorem only applies to right-angled triangles — those with one 90° angle. For triangles without a right angle you need the sine rule or cosine rule instead, which are covered in later trigonometry calculators.
Builders and carpenters use it to check that corners are square; surveyors and navigators use it to measure straight-line distances; architects size diagonal braces and roof rafters; and game developers and computer-graphics engineers use it constantly to compute distances between points on screen.