Solve a right triangle for its hypotenuse or for one unknown leg. The calculation trail makes the reported missing side easier to reproduce.
Legs of 9 and 12 give c² = 81 + 144 = 225, so c = 15. Conversely, a known leg 9 and hypotenuse 15 give the other leg √(225 − 81) = 12. This Pythagorean Theorem example can be compared with coordinate distance.
The theorem checks squareness in construction, finds diagonals, converts coordinate changes to distance, and solves many navigation or layout problems on a flat plane.
In a right triangle, the squares of the two legs add to the square of the hypotenuse: a² + b² = c². The hypotenuse is opposite the right angle and is always the longest side.
A correct missing side therefore depends on choosing the model before entering the numbers.
This relationship only holds for right triangles. When finding a leg, subtract the known leg's square from the hypotenuse's square; reversing them creates an impossible negative radicand. If the Pythagorean Theorem assumptions do not fit, consider missing triangle angle.
Review the sign, scale, and unit of missing side after entering unknown side, known leg and other leg or hypotenuse.
Label the side opposite 90° as c. Square the known lengths, rearrange a² + b² = c² for the unknown, and take the positive square root because a physical length is positive. A hand-worked extension of Pythagorean Theorem is general triangle solving.
Source values may arrive in a different order from the form. Map them explicitly to unknown side, known leg and other leg or hypotenuse, normalize units, and retain enough precision for the next step after missing side.
The hypotenuse grows when either leg grows, but it is less than the sum of the legs. For a missing leg, the result approaches zero as the known leg approaches the hypotenuse. Watching this response separates a data-entry mistake from an unexpected but valid value.
If unknown side, known leg and other leg or hypotenuse are exact counts, more result digits may be meaningful than when they are measured approximations. Let the least certain source guide the final presentation.
The theorem is a side relationship for right triangles. The converse can test rightness: if the longest side squared equals the sum of the other squares, the triangle is right. This page deliberately reports only the former interpretation.
Label the output as missing side in notes or tables. Store unknown side, known leg and other leg or hypotenuse beside it when the result will be reused in a later stage.
It is opposite the right angle and longer than either leg.
No. Use trigonometric laws for general triangles.
Side length represents a positive distance.
A set of positive integers such as 3, 4, and 5 satisfying a² + b² = c².
Yes, as long as the known lengths are positive and geometrically possible.