Pythagorean Theorem Calculator

Calculate missing sides of right triangles using a^2 + b^2 = c^2 with visualization and step-by-step solutions

Check Pythagorean Theorem Calculator

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Common Pythagorean Triples

Click any triple to load it into the calculator

Pythagorean Theorem Formulas

Theorem: a² + b² = c²
Hypotenuse: c = √(a² + b²)
Leg a: a = √(c² - b²)
Leg b: b = √(c² - a²)
Area: A = ½ab
Perimeter: P = a + b + c
Angle A: A = atan(a/b)
Angle B: B = 90° - A

About the Pythagorean Theorem

  • • The theorem only applies to right triangles (one 90° angle)
  • • The hypotenuse is always the longest side, opposite the right angle
  • • Pythagorean triples are integer solutions (e.g., 3-4-5)
  • • Any multiple of a triple is also a triple (e.g., 6-8-10)
  • • The converse: if a² + b² = c², the triangle is right
  • • The distance formula is derived from this theorem
  • • Over 400 different proofs exist for this theorem
  • • Named after Pythagoras (~570-495 BC), though known earlier

Formula result

Copy-ready result handoff

Use this after solving one side of a right triangle.

Common searches
Pythagorean theorem calculatorhypotenuse calculatora squared plus b squared
Input check

Confirm a and b are legs and c is the hypotenuse.

Assumption

The theorem applies only to right triangles.

Pythagorean result note

Copy this after replacing the bracketed result with the value shown by the calculator.

Pythagorean result: [paste known sides, missing side, and formula steps].
Inputs checked: Confirm a and b are legs and c is the hypotenuse.
Assumption: The theorem applies only to right triangles.
Use case: Geometry homework, distance check, construction measure, or diagonal estimate.
Next check: Use Right Triangle Calculator when angles, area, and perimeter are also needed.

Enter two right-triangle sides to solve the third, or enter three side lengths to test whether they form a right triangle. Keep the longest side as c in verify mode.

How the result is built

Formula, Variables, And Worked Example

a^2 + b^2 = c^2

a and b are perpendicular legs. c is the hypotenuse opposite the right angle and must be the longest side.

Worked example

With legs 9 and 12, c = sqrt(9^2 + 12^2) = sqrt(225) = 15. The triangle is a scaled 3-4-5 right triangle.

Assumptions and common mistakes
  • The theorem applies only to right triangles or to testing whether a triangle is right.
  • Do not put a leg value in the hypotenuse position when solving for a missing leg.
  • When testing three sides, sort them first and use the longest side as c.
  • Lengths must use the same unit before squaring and adding.
Mode guide
  • Find hypotenuse: use two known perpendicular legs.
  • Find leg: use one leg and the hypotenuse.
  • Verify: use three side lengths, with the longest entered as c.
Related checks
Right triangleDistanceTriangleSlope

Pythagorean Formula, Examples, And Mistakes

How to choose the right solve mode

Use hypotenuse mode when the two perpendicular legs are known. Use leg mode when one leg and the hypotenuse are known. Use verify mode when all three side lengths are known and you need to test whether the shape can be a right triangle.

What the result includes

The calculator returns the missing side, the two acute angles, area, perimeter, and a labeled triangle diagram. The step-by-step section shows the squared values before taking the square root, which helps catch side placement mistakes.

Pythagorean triples and quick checks

Integer triples such as 3-4-5, 5-12-13, and 8-15-17 are common right triangles. Any scaled version of a triple is also a right triangle, so 6-8-10 has the same shape as 3-4-5.

Coordinate distance connection

The distance formula is the Pythagorean theorem applied to horizontal and vertical coordinate differences. Use the Distance Calculator for point-to-point geometry, or the Right Triangle Calculator when angle and altitude outputs are also needed.

References & Sources

Common result checks

Questions about this tool

Which side is the hypotenuse?
The hypotenuse is the side opposite the 90 degree angle. It is always the longest side in a right triangle.
Can the Pythagorean theorem solve any triangle?
No. It applies to right triangles. For non-right triangles, use a general triangle solver with the law of sines or law of cosines.
How do I verify whether three sides make a right triangle?
Put the longest side in the c position. If a^2 + b^2 equals c^2, the triangle is right.
Do all side lengths need the same unit?
Yes. Convert all lengths to the same unit before squaring, adding, or comparing sides.