What is the Pythagorean theorem?

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle, denoted as 'c') is equal to the sum of the squares of the lengths of the other two sides (denoted as 'a' and 'b'). The formula is a² + b² = c².

What is a hypotenuse?

The hypotenuse is the longest side of a right-angled triangle. It is always the side opposite the 90-degree angle.

Can this theorem be used for any triangle?

No. The Pythagorean theorem only applies to right-angled triangles, which are triangles that have one angle that is exactly 90 degrees.

Pythagorean Theorem Calculator (a² + b² = c²)

How Does This Calculator Work?

This calculator is based on the Pythagorean theorem, a fundamental principle in geometry that describes the relationship between the three sides of a right-angled triangle.

a² + b² = c²

Here, 'a' and 'b' are the lengths of the two shorter sides (legs) of the triangle, and 'c' is the length of the hypotenuse (the longest side, opposite the right angle).
To find any missing side, the calculator rearranges this formula:
- To find c: c = √(a² + b²)
- To find a: a = √(c² - b²)
- To find b: b = √(c² - a²)

The Surprising History of the Pythagorean Theorem

While the theorem is named after the ancient Greek mathematician Pythagoras (c. 570 – c. 495 BC), the relationship it describes was known to other civilizations long before him. Clay tablets from ancient Babylon, dating back to 1800 BC, show that they had knowledge of "Pythagorean triples"—sets of integers like (3, 4, 5) that fit the formula.

So why does Pythagoras get the credit? The Pythagorean school was one of the first to move from simply observing this relationship to creating a formal, logical proof that it was true for all right-angled triangles. This shift from practical observation to abstract proof was a critical step in the development of modern mathematics. The theorem is not just about triangles; it's a statement about the fundamental nature of space itself.

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Frequently Asked Questions (FAQ)

What is a right-angled triangle?

A right-angled triangle (or right triangle) is a triangle in which one of the three angles is exactly 90 degrees. The Pythagorean theorem only works for this type of triangle.

How do I know which side is the hypotenuse (c)?

The hypotenuse is always the longest side of a right triangle. It is located directly opposite the 90-degree right angle.

What happens if I enter a side 'a' or 'b' that is longer than the hypotenuse 'c'?

This is a geometric impossibility. If you are solving for a missing side (a or b) and enter a value for the other side that is greater than or equal to the hypotenuse, the calculator will show an error. In a right triangle, the hypotenuse must always be the longest side.

Pythagorean Theorem Calculator

The length of the missing side is: