Find the hypotenuse
When both legs are known, add their squares and take the square root to find the longest side.
c = √(3² + 4²)
Result: c = 5
Pythagorean Theorem Calculator
The Pythagorean Theorem calculator helps you find the length of any side of a right triangle. Just enter the lengths of two sides, and we'll calculate the third side for you.
Calculate Triangle Sides
Select which side you want to calculate, then enter the values for the other two sides.
Solve for:
What is the Pythagorean Theorem?
The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of squares of the other two sides. It's one of the most fundamental relationships in Euclidean geometry.
The Formula
a² + b² = c²
If a and b are the legs of a right triangle and c is the hypotenuse, then a² + b² = c². This means we can find any side if we know the other two.
Historical Background
Named after the ancient Greek mathematician Pythagoras (570-495 BCE), this theorem was known to ancient civilizations including Babylonians and Egyptians. While Pythagoras is credited with the first proof, the concept has been discovered independently by multiple cultures throughout history.
How to Use the Pythagorean Theorem
- Identify the right triangle and label the sides: the two legs as a and b, and the hypotenuse (the side opposite to the right angle) as c.
- Determine which side you need to find (a, b, or c).
- Use the appropriate form of the equation: c = √(a² + b²) to find the hypotenuse, or a = √(c² - b²) or b = √(c² - a²) to find a leg.
- Substitute the known values and solve for the unknown side.
Make sure the triangle is a right triangle before applying the theorem. If you are solving for a leg, the hypotenuse must be larger than the other known side.
Real-World Applications
- Construction and carpentry: Ensuring walls are perpendicular to floors or measuring diagonals.
- Navigation: Calculating the shortest distance between two points.
- Physics: Resolving forces and vectors in mechanics.
Important Notes
- The Pythagorean Theorem only works for right triangles (triangles with one 90° angle).
- The hypotenuse is always the longest side of the right triangle and is opposite to the right angle.
Pythagorean Theorem Examples
Find a missing leg
When the hypotenuse and one leg are known, subtract squares before taking the square root.
a = √(13² - 5²)
Result: a = 12
Use a common right-triangle triple
Many geometry problems use familiar triples to verify side lengths quickly.
5² + 12² = 13²
Result: 25 + 144 = 169
Pythagorean Theorem FAQ
When can I use the Pythagorean theorem?
You can use it only for right triangles, where one angle measures exactly 90 degrees.
Which side is the hypotenuse?
The hypotenuse is the side opposite the right angle, and it is always the longest side of the triangle.
Why does the calculator reject some inputs when solving for a leg?
If the known hypotenuse is not longer than the other known side, the triangle is not valid for the theorem and the square root step would fail.
Can this theorem be used for non-right triangles?
No. For non-right triangles you need other formulas such as the law of cosines or the law of sines.