Pythagorean Theorem Calculator

Calculate Missing Sides of Right Triangles

a (leg) b (leg) c (hypotenuse)
a² + b² = c²

Enter any two values to calculate the third

Result:

Understanding the Pythagorean Theorem

The Pythagorean theorem is one of the most famous and useful theorems in mathematics, discovered by the ancient Greek mathematician Pythagoras around 500 BCE. It describes the fundamental relationship between the three sides of a right triangle, making it possible to calculate any missing side when the other two are known.

The theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Expressed as a formula: a² + b² = c², where c represents the hypotenuse and a and b represent the two legs of the triangle.

Practical Applications

This simple yet powerful relationship has countless real-world applications. Architects and construction workers use it to ensure buildings are square and to calculate diagonal bracing. Surveyors use it to measure distances that would be difficult to measure directly. Navigation systems use it to calculate the shortest distance between two points.

In computer graphics and game development, the Pythagorean theorem calculates distances between objects on screen. Engineers use it in structural analysis and design. Even smartphone GPS systems rely on variations of this theorem to pinpoint your location. Understanding this theorem opens doors to solving practical problems in numerous fields.

Frequently Asked Questions

What is the Pythagorean theorem?
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (longest side) equals the sum of squares of the other two sides. The formula is a² + b² = c², where c is the hypotenuse and a and b are the other two sides.
How do you find the hypotenuse?
To find the hypotenuse (c), use the formula c = √(a² + b²). Square both legs, add them together, then take the square root of the sum. For example, if a=3 and b=4, then c = √(9 + 16) = √25 = 5.
How do you find a missing leg?
To find a missing leg, use a = √(c² - b²) or b = √(c² - a²). Square the hypotenuse, subtract the square of the known leg, then take the square root. For example, if c=5 and a=3, then b = √(25 - 9) = √16 = 4.
What is a right triangle?
A right triangle is a triangle with one 90-degree angle. The side opposite the right angle is called the hypotenuse and is always the longest side. The other two sides are called legs. The Pythagorean theorem only works for right triangles.
What are practical uses of the Pythagorean theorem?
The Pythagorean theorem is used in construction (ensuring corners are square), navigation (calculating shortest distances), surveying, architecture, computer graphics, physics (calculating vectors), and many engineering applications. It's fundamental in any field dealing with distances and right angles.