Square both legs and add them, then take the square root for the hypotenuse, or subtract a squared leg from the squared hypotenuse and take the square root for a missing leg.
Pythagorean Theorem Calculator
Find the hypotenuse of a right triangle from its two legs, or a missing leg from the hypotenuse and the other leg.
Calculating Hypotenuse and Missing Legs with the Pythagorean Theorem
The Pythagorean theorem relates the three sides of any right triangle, letting you solve for the hypotenuse when both legs are known, or for a missing leg when the hypotenuse and one leg are known. This calculator handles both directions, covering searches like Pythagorean theorem calculator, hypotenuse calculator, and right triangle side calculator.
Related calculators you may need next: Right Triangle Area Calculator and Triangle Solver.
How the Pythagorean Theorem Calculator Works
Hypotenuse Formula
c = √(a² + b²)
Hypotenuse Worked Example
Legs a=6 and b=8: c = √(6² + 8²) = √(36+64) = √100 = 10.
Missing Leg Formula
a = √(c² − b²)
Missing Leg Worked Example
Hypotenuse c=13 and leg b=5: a = √(13² − 5²) = √(169−25) = √144 = 12.
Why This Theorem Only Applies to Right Triangles
The relationship a² + b² = c² holds specifically because of the 90° angle between the two legs — for any other triangle shape, this exact relationship no longer holds, and a different approach (like the law of cosines) is needed to relate the sides.
Related: Triangle Slope / Grade / Pitch Calculator.
Frequently Asked Questions
Which side is the hypotenuse?
The hypotenuse is always the longest side of a right triangle, located directly opposite the 90° angle — the other two sides, called legs, form the right angle itself.
Can I use this calculator if I only know one side?
No — you need at least two of the three sides (either both legs, or the hypotenuse plus one leg) to solve for the remaining unknown side using this theorem.
What are Pythagorean triples, and why do they come up often?
Pythagorean triples are sets of three whole numbers that satisfy a²+b²=c² exactly, like 3-4-5 or 6-8-10 — they appear often in examples and textbook problems because they produce clean, whole-number answers.
Does this calculator work with decimal or non-whole-number side lengths?
Yes — the Pythagorean theorem applies to any positive real-number side lengths, not just whole numbers, so decimal measurements work exactly the same way through the same formula.
What real-world situations use the Pythagorean theorem?
Common applications include finding the diagonal distance across a rectangular space, determining the correct length of a ladder or ramp, and calculating straight-line distance between two points using their horizontal and vertical separation as the two legs.
Why must the value under the square root be positive?
Since side lengths are always positive real numbers, c²−b² must be positive for a valid missing-leg calculation — if b is entered larger than c, the result would require a square root of a negative number, which signals invalid input rather than a valid triangle.
Is the Pythagorean theorem the same as the distance formula?
Yes — the distance formula between two coordinate points is derived directly from the Pythagorean theorem, treating the horizontal and vertical differences between the points as the two legs of a right triangle.
