What is the Pythagorean Theorem?

The Pythagorean Theorem is a fundamental principle in Euclidean geometry that describes the relationship between the three sides of a right-angled triangle. A right-angled triangle is any triangle that has one angle equal to exactly 90 degrees. The theorem states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs).

The Formula: a² + b² = c²

In this famous equation:

• a and b are the lengths of the two legs of the right triangle.
• c is the length of the hypotenuse.

How to Solve for Each Side

By rearranging the formula, you can solve for any of the three sides if you know the other two:

• To find the hypotenuse (c): c = √(a² + b²)
• To find side a: a = √(c² - b²)
• To find side b: b = √(c² - a²)

A key rule is that the hypotenuse (c) must always be the longest side. Our calculator validates this to prevent impossible calculations.

Real-World Applications of the Theorem

The Pythagorean theorem is not just an academic exercise; it's used extensively in the real world:

• Construction and Architecture: Ensuring that corners of buildings are perfectly square (90 degrees). A builder might measure 3 units along one wall and 4 units along the adjacent wall; the distance between those two points should be exactly 5 units for a perfect right angle.
• Navigation: Calculating the shortest distance between two points. The distance formula, which is used in GPS and mapping, is a direct application of this theorem. See it in action with our Distance Calculator.
• Surveying: Calculating the slope of a piece of land or the height of a hill. You can also calculate slope directly with our Slope Calculator.
• Design: From graphic design layouts to engineering blueprints, the theorem helps in creating precise geometric shapes and structures.

For more tools related to geometry, explore our full suite of Math calculators.