How the Pythagorean Theorem Works

One right angle, one identity

Label the legs a and b and the side opposite the right angle c. Then a² + b² = c². Legs 3 and 4 give 9 + 16 = 25, so c = 5. The Pythagorean theorem calculator is that identity in two directions: hypotenuse from both legs, or a missing leg from the hypotenuse and the other leg. Units stay whatever you typed — centimetres stay centimetres.

The hypotenuse must be the longest side. If you ask for a leg and type a “hypotenuse” shorter than the known leg, there is no real triangle. Equal values would force the other leg to 0.

Finding a leg, not only the long side

Rearrange: a = √(c² − b²). Hypotenuse 13 and known leg 5 give √(169 − 25) = 12. That is the 5-12-13 triple. The same algebra is a distance: from (0, 0) to (5, 12) is 13. When your inputs are coordinates rather than named legs, use the distance calculator. Slope is a different ratio (rise over run) and does not replace c.

The converse, and triangles that are not right

If you already have three sides and a² + b² equals c², the angle opposite c is 90°. If the sum is larger than c² the angle is acute; if smaller, obtuse. Pythagoras does not compute those other cases. Use the triangle calculator for SSS, SAS or ASA that are not right, and the right-triangle calculator when you also want the two acute angles, area and perimeter from the same right triangle.

3-4-5 is a family, not a spell

Any positive scale of 3-4-5 works: 6-8-10, 9-12-15, 1.5-2-2.5. Other integer triples exist (5-12-13, 8-15-17). Most real measurements are not integers. Rounding each side independently can make a true right triangle fail a sloppy “does 3-4-5 fit?” check — compute the squares instead of hunting a famous triple.