pythagorean theorem formula calculator

What Is the Pythagorean Theorem?

The Pythagorean theorem is one of the most important formulas in geometry. It applies to right triangles, which are triangles with one 90° angle. The theorem states that the square of the hypotenuse equals the sum of the squares of the two legs.

Written as a formula: a² + b² = c²

• a and b are the legs (the two shorter sides that meet at the right angle).
• c is the hypotenuse (the longest side, opposite the right angle).

How to Use This Calculator

Step-by-step

• Enter two known side lengths.
• Leave the unknown side blank.
• Click Calculate.
• The calculator fills in the missing side and shows the equation used.

This works for both situations:

• Finding the hypotenuse: c = √(a² + b²)
• Finding a leg: a = √(c² − b²) or b = √(c² − a²)

Examples

Example 1: Find the hypotenuse

If a = 3 and b = 4:
c = √(3² + 4²) = √(9 + 16) = √25 = 5

Example 2: Find a leg

If c = 13 and b = 5:
a = √(13² − 5²) = √(169 − 25) = √144 = 12

Common Mistakes to Avoid

• Using the theorem on triangles that are not right triangles.
• Mixing up the hypotenuse with a leg (the hypotenuse is always the longest side).
• Using negative or zero side lengths.
• Trying to calculate a leg when c is not larger than the other known leg.

Real-Life Uses

The Pythagorean theorem appears in many practical situations, including:

• Construction and carpentry (checking square corners).
• Navigation and map distance calculations.
• Computer graphics and game development.
• Physics and engineering problem solving.

Quick FAQ

Can I use decimals?

Yes. The calculator accepts whole numbers and decimals.

What if I enter all three sides?

The calculator will verify whether your values satisfy the theorem and therefore represent a right triangle.

Do units matter?

Any unit works (meters, feet, inches) as long as all sides use the same unit.