MultiConvers

Pythagorean Theorem Calculator

Apply a² + b² = c² and nothing else: recover a hypotenuse from the two legs, or a missing leg from the hypotenuse and the other leg, with no angles and no area.

How to use

  1. Choose whether you are finding the hypotenuse from legs a and b, or finding a leg from hypotenuse c and the other known leg.
  2. Enter the two lengths you have. They must be positive and already in the same unit; the missing side is reported in that unit.
  3. If you are solving for a leg, confirm c is strictly longer than the known leg. Equal lengths would force a zero side, which is rejected.
  4. Stop after the one missing length. For complementary angles, ½ab area or the perimeter, open the right-triangle calculator instead.

How it works

The theorem is a statement about squares on the sides of a right triangle: the square built on the hypotenuse equals the sum of the squares on the legs. Algebraically that is a² + b² = c². Solving for the long side is c = √(a² + b²). Solving for a missing leg is b = √(c² − a²). Legs 3 and 4 produce 5; hypotenuse 5 with a known leg of 3 produces 4. Those are lengths, not degrees and not unit².

A ladder against a wall, a guy-wire to the top of a pole, the diagonal of a rectangular gate and the throw from home plate to second on a square diamond are the classic jobs. You care about the missing length and you do not need the acute angles. If you later want those angles or the enclosed area, you have left this page’s scope.

This is not trigonometry. There is no sine, cosine or arctan here, and no law of cosines for a triangle that is not right-angled. An obtuse 3-4-6 satisfies the triangle inequality but not a² + b² = c², so it is not a Pythagorean triple and this calculator will not pretend 6 is a hypotenuse of 3 and 4. The general triangle page will still accept 3-4-6 and report its actual angles.

It is also not the distance calculator, though the arithmetic is the same idea: the gap from (0, 0) to (3, 4) is √(3² + 4²) = 5. Use coordinates when the given data are points; use this page when the given data are already named legs and a hypotenuse.

Common integer families — 3-4-5, 5-12-13, 8-15-17 and their multiples — are a quick mental check. Any similar scaling (6-8-10, 9-12-15) still obeys the same identity. A hypotenuse shorter than or equal to a typed “leg” is geometrically impossible and returns no result rather than an imaginary length.

Formula

a² + b² = c²
c = √(a² + b²)
b = √(c² − a²)

Worked examples

  • Find the hypotenuse

    legs 3 and 4 → 5

  • Find a leg

    hypotenuse 5, other leg 3 → 4

Useful notes

  • Only right triangles. An obtuse 3-4-6 is not a Pythagorean triple.
  • No angles, area or perimeter — those extras are on the right-triangle page.
  • Finding a leg requires hypotenuse > known leg; otherwise the radicand is not positive.
  • Integer families such as 3-4-5 and 5-12-13 are checks, not a separate mode.

FAQ

What is the Pythagorean theorem?

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: a² + b² = c².

Why was my missing-leg calculation rejected?

The hypotenuse must be longer than the known leg. If they are equal, the other leg would be 0, which is not a triangle.

Does this give the angles too?

No. Use the right-triangle calculator when you also want angles, area and perimeter.

Can I use this on sides 3, 4 and 6?

Not as a Pythagorean triple. 3² + 4² is 25, not 36. Those lengths can still form an obtuse triangle on the triangle calculator.

Is the segment from (0, 0) to (3, 4) the same arithmetic?

Yes, √(3² + 4²) = 5. When your inputs are coordinates rather than named legs, use the distance calculator.

Guides

Related tools

See all Geometry tools.