Use this Pythagorean theorem calculator to find a missing side of a right triangle or check whether three sides fit the theorem.
How to use our Pythagorean Theorem Calculator
- Choose an option in What do you want to do: find the hypotenuse, find leg a, find leg b, or check three sides.
- Pick one Length unit and use that same unit for every side you enter.
- If you are finding the hypotenuse, enter values for Side a (selected unit) and Side b (selected unit).
- If you are finding a leg, enter the known leg and Hypotenuse c (selected unit). Make sure the hypotenuse is longer than the known leg.
- If you are using check mode, enter Side a (selected unit), Side b (selected unit), and Hypotenuse c (selected unit). Then choose Check tolerance if your numbers were rounded.
- Click Calculate to see the result, the Equation used, and any check values shown for your mode.
- Sanity-check the answer: a computed Missing side should be positive, and in check mode the Difference should be very close to 0 if the sides make a right triangle.
Definitions
Right triangle: A triangle with one 90 degree angle.
Leg: One of the two shorter sides that meet to form the right angle.
Hypotenuse c: The longest side of a right triangle, opposite the right angle [2][3].
Missing side: The side length the calculator solves for from the other two known sides.
Equation used: The version of the Pythagorean theorem used for your selected mode.
Check tolerance: The allowed difference when deciding whether rounded side lengths still count as a right triangle.
Difference: The value of a^2 + b^2 - c^2. A result near 0 means the sides fit the theorem well.
Common mistakes and quick fixes
Mistake: Using different units for different sides, such as feet for Side a (selected unit) and inches for Hypotenuse c (selected unit) .
Fix: Choose one Length unit and convert all side lengths to that same unit before calculating.
Mistake: Entering a shorter value in Hypotenuse c (selected unit) than in a known leg when solving for a leg.
Fix: The hypotenuse must be the longest side, so enter a Hypotenuse c (selected unit) value greater than the known leg.
Mistake: Typing 0 or a negative number into Side a (selected unit) , Side b (selected unit) , or Hypotenuse c (selected unit) .
Fix: Use positive side lengths only. A right triangle cannot have a side length of 0 or less.
Mistake: In check mode, putting the largest side into Side a (selected unit) or Side b (selected unit) instead of Hypotenuse c (selected unit) .
Fix: Put the longest side in Hypotenuse c (selected unit) . The tool does not swap sides automatically.
Mistake: Expecting rounded decimal sides to make the Difference exactly 0 in check mode.
Fix: Adjust Check tolerance to match how much rounding your side lengths use, such as hundredth or tenth.
Mistake: Reading a^2 + b^2 or c^2 as side lengths instead of squared values.
Fix: Treat a^2 + b^2 , c^2 , and Difference as comparison values for checking the theorem, not as regular side lengths.
Limitations & Key Assumptions / Boundary Conditions
- This tool is only for right triangles, where the sides follow a^2 + b^2 = c^2.
- All entered side lengths must use the same Length unit. The calculator cannot detect if you mixed units mentally.
- When solving for a leg, Hypotenuse c (selected unit) must be longer than the known leg or there is no real right triangle with those values.
- In check mode, the calculator assumes the value entered as Hypotenuse c (selected unit) is the longest side and does not reorder the sides for you.
- Rounded decimals may fail an exact check even when the original triangle was right, so the result can depend on the chosen Check tolerance.
- Very large or heavily rounded numbers can make the displayed values easier to read than to use for exact hand-checking.
Methodology
Core idea
The calculator uses the Pythagorean theorem for right triangles: the sum of the squares of the two legs equals the square of the hypotenuse [3].
a^2 + b^2 = c^2
How each mode works
If you choose to find the hypotenuse, the calculator squares both legs, adds them, and takes the square root.
c = sqrt(a^2 + b^2)
If you choose to find leg a, it subtracts the square of leg b from the square of the hypotenuse, then takes the square root.
a = sqrt(c^2 - b^2)
If you choose to find leg b, it subtracts the square of leg a from the square of the hypotenuse, then takes the square root.
b = sqrt(c^2 - a^2)
In check mode, it compares the left side of the theorem with the right side.
difference = (a^2 + b^2) - c^2
If the absolute value of difference is less than or equal to the chosen tolerance, the result says the sides make a right triangle within that tolerance.
abs(difference) <= tolerance
Worked mini-example
Example: if Side a is 3 and Side b is 4, then the calculator computes 3^2 = 9 and 4^2 = 16, adds them to get 25, and takes the square root. The Missing side is 5.
c = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5
For a check example, if a = 5, b = 12, and c = 13, then 5^2 + 12^2 = 25 + 144 = 169 and 13^2 = 169, so the difference is 0 and the triangle passes the check.
difference = (5^2 + 12^2) - 13^2 = 169 - 169 = 0
Validation used
The calculator blocks side lengths that are 0 or negative. It also blocks impossible leg-solving cases where the entered hypotenuse is not longer than the known leg, because that would make the value inside the square root negative. In check mode, it requires Hypotenuse c to be the largest entered side.
Interpretation notes
A positive Difference means a^2 + b^2 is larger than c^2. A negative Difference means c^2 is larger. The closer the value is to 0, the closer the three sides are to fitting a right triangle.
Assumptions behind the result
This method assumes a flat, ordinary geometric right triangle and consistent units for all side lengths. Real-world measurements may differ slightly because of rounding, measuring error, or copied homework values with limited decimal places.