Pythagorean Theorem Calculator

Understanding the Pythagorean Theorem

The Pythagorean Theorem is one of the most famous and widely applied formulas in geometry and mathematics. It provides a simple relationship between the three sides of a right-angled triangle. Whether you are building a staircase, calculating the distance of a shortcut, or studying for a math exam, understanding how to use a² + b² = c² is an essential skill. Our free online calculator makes it incredibly easy to find any missing side without having to do the algebraic heavy lifting yourself.

The Formula: a² + b² = c²

In a right-angled triangle, there are two shorter sides (called the legs) forming the 90-degree angle, and one long side across from the right angle (called the hypotenuse).

• a = Length of the first leg
• b = Length of the second leg
• c = Length of the hypotenuse (the longest side)

The theorem states that if you square the lengths of the two legs and add them together, the result will exactly equal the square of the hypotenuse.

How to Use the Pythagorean Calculator

While the base formula is always a² + b² = c², you might not always be looking for the hypotenuse. Sometimes you know the hypotenuse and one leg, and you need to find the other leg. Our calculator adapts to your needs:

1. Solving for the Hypotenuse (c)

If you know legs a and b, you use the standard formula. The calculator will perform:
c = √(a² + b²)

2. Solving for Leg (a)

If you know the hypotenuse c and leg b, you rearrange the formula to isolate a. The calculator will perform:
a = √(c² - b²)

3. Solving for Leg (b)

Similarly, if you know the hypotenuse c and leg a, you isolate b. The calculator will perform:
b = √(c² - a²)

Real-World Applications

The Pythagorean Theorem is not just an abstract mathematical concept; it is used daily in various professions:

• Architecture & Construction: Builders use the 3-4-5 rule (a Pythagorean triple) to ensure walls are perfectly square (90 degrees) to one another.
• Navigation: If a ship travels 30 miles North and 40 miles East, the captain can use the theorem to determine their exact straight-line distance (50 miles) from the starting point.
• Surveying: Surveyors use it to calculate the steepness of slopes of hills and mountains.

Example Calculation

Imagine you have a ladder leaning against a wall. The base of the ladder is 5 feet away from the wall (a = 5), and it reaches a height of 12 feet up the wall (b = 12). How long is the ladder (c)?

Step 1: Square both legs: 5² = 25, and 12² = 144.
Step 2: Add them together: 25 + 144 = 169.
Step 3: Take the square root of the sum: √169 = 13.

Therefore, the ladder must be exactly 13 feet long. Our calculator performs these exact steps and displays them for you so you can follow along with the logic.