Pythagorean Theorem Calculator

Find the missing side of a right triangle or verify if three sides form a right triangle.

Pythagorean Theorem Calculator tool

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Example

Input: Find the hypotenuse: side A = 3, side B = 4

Result: Hypotenuse C = 5 (Pythagorean triple: Yes; angles 36.8699 / 53.1301 / 90 degrees)

c = sqrt(a^2 + b^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.

Common Right Triangles

Legs a, bHypotenuse cAcute angles (degrees)Area
3, 4536.8699 / 53.13016
5, 121322.6199 / 67.380130
8, 151728.0725 / 61.927560
7, 242516.2602 / 73.739884
20, 212943.6028 / 46.3972210
9, 404112.6804 / 77.3196180
6, 81036.8699 / 53.130124
1, 11.41421445 / 450.5

All but the last row are whole-number Pythagorean triples.

  1. Select the mode: Find Hypotenuse, Find Side A, Find Side B, or Verify Triangle.
  2. In a find mode, enter the two known sides (the mode sets which side is calculated).
  3. View the missing side length calculated using the Pythagorean theorem, plus the area, perimeter, and angles.
  4. In verify mode, enter all three sides to check if they satisfy a^2 + b^2 = c^2.
  5. Compare the sum of the squared legs with the squared hypotenuse and the difference shown in verify mode.
  6. Click the copy button to copy the result.

Using a^2 + b^2 = c^2 in Real Life

The Pythagorean theorem says that in a right triangle, a^2 + b^2 = c^2, where c is the hypotenuse, the side opposite the right angle. With legs of 3 and 4, the hypotenuse is 5. This calculator finds any missing side from the other two, or checks whether three sides make a right triangle.

Pick a mode first. Find Hypotenuse uses c = sqrt(a^2 + b^2). Find Side A and Find Side B work backward from the hypotenuse and the other leg, using a = sqrt(c^2 - b^2). With the missing side, you also get the area, the perimeter and both acute angles. You also see whether the three sides form a whole-number Pythagorean triple.

The theorem turns up in all kinds of practical jobs. To check that a corner is square, measure 3 feet along one wall and 4 feet along the other. The diagonal between the marks should be exactly 5 feet. A 16:9 TV that is 48 inches wide and 27 inches tall has a 55.07-inch diagonal. A 5 m ladder with its foot 1.5 m from a wall reaches 4.77 m up.

Verify Triangle mode takes all three sides. It treats the longest one as the hypotenuse, whichever box you typed it in. It compares the sum of the squared legs with the squared hypotenuse and shows the difference. A tiny gap from rounding is allowed, so 1, 1 and 1.414214 still counts as right. If the sides are not a right triangle, it tells you whether the triangle is acute, obtuse, or impossible. For triangles without a right angle, use the Triangle Calculator. To take other roots, try the Square Root and Nth Root Calculator.

Formula / How It Works

a^2 + b^2 = c^2 | c = sqrt(a^2 + b^2) | a = sqrt(c^2 - b^2)

Frequently Asked Questions

In any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a^2 + b^2 = c^2. The hypotenuse is the longest side, opposite the right angle. The rule only holds for right triangles, which is why it can also test whether a corner is square.
Square both legs, add them, and take the square root. With legs of 5 and 12: 25 + 144 = 169, and the square root of 169 is 13. In the calculator, choose Find Hypotenuse and enter the two legs as Side A and Side B.
Square the hypotenuse, subtract the square of the known leg, and take the square root. With a hypotenuse of 10 and a leg of 6: 100 - 36 = 64, so the missing leg is 8. Choose Find Side A or Find Side B, then enter the known leg and the hypotenuse. The hypotenuse must be the longer of the two.
In the three find modes, yes: Hypotenuse C is where the long side goes. In Verify Triangle mode it does not matter. The calculator sorts the three sides and treats the longest one as the hypotenuse, wherever you typed it.
They are sets of three whole numbers that fit a^2 + b^2 = c^2 exactly. Common ones are 3-4-5, 5-12-13, 8-15-17 and 7-24-25. Any multiple of a triple also works, so 6-8-10 and 9-12-15 are triples too. The calculator tells you when your sides form one.

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