Pythagorean theorem calculator

The Pythagorean theorem says that in a right triangle the two shorter sides, the legs a and b, and the longest side, the hypotenuse c, satisfy a² + b² = c². To find the hypotenuse, square the legs, add, and take the square root. With legs of 5 and 12: 5² + 12² = 25 + 144 = 169, and √169 = 13.

Here is one of the many proofs, and the quickest to see. Draw a square with sides a + b and fit four copies of the right triangle inside it, one in each corner, with their hypotenuses facing inwards. The hypotenuses enclose a tilted square of side c. The big square's area is (a + b)², and it is also the inner square plus the four triangles: c² + 4 × ½ab. Expanding (a + b)² gives a² + 2ab + b², the 2ab cancels against the four triangles, and a² + b² = c² is left. With 5 and 12: the outer square is 17² = 289, the triangles are 4 × 30 = 120, and 289 - 120 = 169 = 13².

Solving for a leg runs it the other way: subtract instead of adding. With a hypotenuse of 13 and a leg of 12, the other leg is √(169 - 144) = √25 = 5. In this calculator, enter c and the leg you know and leave the other leg at 0. Adding when you should subtract is the common slip, and it gives itself away: the "leg" comes out longer than the hypotenuse, which no right triangle has. The other mistake is labeling. c must be the side opposite the right angle; plug the hypotenuse in as a leg and the answer is wrong without looking wrong.

Every side must be in the same unit, and the answer comes out in that unit. The squares in the middle of the calculation are areas (25 and 144 square units), but the square root brings them back to a length.

The theorem only holds for right triangles. Its converse is just as useful: if three sides satisfy a² + b² = c², the angle opposite c is exactly 90°. That is how builders check a corner: measure 3 feet along one wall and 4 along the other, and if the diagonal is 5 feet, the corner is square. For a triangle without a right angle, the law of cosines adds a correction term, c² = a² + b² - 2ab cos C, which this calculator's other modes use.

5, 12, 13 is a Pythagorean triple, three whole numbers that fit; most pairs of legs do not give one. Legs of 5 and 7 give √74, about 8.602, and the page shows that exact radical as well as the decimal. The same right triangle here has an area of ½ × 5 × 12 = 30 and acute angles of about 22.62° and 67.38°.

The theorem turns up wherever a straight-line distance is wanted. The distance formula between two points is it, with the legs being the differences in x and in y. In three dimensions it extends to √(a² + b² + c²), the diagonal of a box. And a TV's screen size is the diagonal of a rectangle, which is the hypotenuse of the triangle its width and height make.