Pythagorean Theorem Calculator

The Pythagorean theorem is arguably the most famous rule in geometry. It provides a simple, direct way to calculate the length of one side of a right-angled triangle if you know the lengths of the other two sides.

Our free online Pythagorean Theorem Calculator is designed to instantly solve these geometric problems. Whether you are an engineering student in Delhi, an architect drafting building dimensions, a carpenter checking if a frame is perfectly square, or just trying to find the shortest distance across a rectangular field, this tool provides perfect accuracy.

You no longer have to manually calculate squares and square roots. Designed for everyday users in India, simply input your known dimensions, and the calculator will find your missing hypotenuse or leg length in a fraction of a second.

How to Use the Calculator

Our interface is built to be visual and straightforward. To find your missing side, just follow these simple steps:

Select the side to find: Use the dropdown menu labeled "Which side do you want to find?". You can choose to calculate the Hypotenuse (c), Leg (a), or Leg (b).
Input known values: Based on your selection, input fields will appear for the remaining two known sides. For instance, if you are finding the hypotenuse, simply enter the lengths for Side A and Side B.
View instant results: As you type, the calculator instantly computes the math. Your missing side's length will be highlighted in the main result box.
Review the step-by-step solution: Look directly beneath the result to see the full equation. The calculator displays exactly how the squares were added or subtracted and how the final square root was applied, making it an excellent study aid for math homework!

Understanding the Formulas

The theorem states that in a right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.

The Core Formula

a² + b² = c²

To calculate a specific missing side, we rearrange this primary equation:

Finding the Hypotenuse (c): If you know both shorter legs, square them, add them together, and find the square root.
c = √(a² + b²)
Finding Leg (a): If you know the hypotenuse and leg b, square both, subtract b² from c², and find the square root.
a = √(c² - b²)
Finding Leg (b): Same logic as above, but subtracting a² from c².
b = √(c² - a²)

Real-Life Worked Examples

Example 1: The Ladder Problem

Scenario: You need to reach a window that is exactly 4 meters (a) above the ground. To keep the ladder stable, the base must be placed exactly 3 meters (b) away from the wall. How long must the ladder (the hypotenuse) be?

Formula: c = √(a² + b²)
Square the sides: 4² = 16, and 3² = 9
Add them: 16 + 9 = 25
Square root: √25 = 5

Result: The ladder must be exactly 5 meters long (c = 5).

Example 2: Buying a Television

Scenario: TV sizes are measured diagonally (the hypotenuse). You are looking at a 50-inch TV (c). You measure the height of your cabinet space, and the TV's height is 30 inches (a). How wide is the TV (b) to ensure it fits in your living room?

Formula: b = √(c² - a²)
Square the sides: 50² = 2500, and 30² = 900
Subtract height from diagonal: 2500 - 900 = 1600
Square root: √1600 = 40

Result: The width of the television is 40 inches (b = 40).

Frequently Asked Questions

What is the Pythagorean Theorem?▼

The Pythagorean theorem is a fundamental rule in geometry that describes the mathematical relationship between the three sides of a right-angled triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Does this theorem work on all triangles?▼

No. The Pythagorean theorem only applies to right-angled triangles—triangles where one of the inside angles is exactly 90 degrees.

What is the hypotenuse?▼

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

What is a Pythagorean triple?▼

A Pythagorean triple is a set of three positive whole numbers (integers) that perfectly fit the a² + b² = c² formula. A common example used in construction is the 3-4-5 triangle (3² + 4² = 5²).

How do I find a missing leg (a or b) instead of the hypotenuse?▼

To find a missing leg, you rearrange the formula. You subtract the square of the known leg from the square of the hypotenuse, and then find the square root of that result. The formula is a = √(c² - b²).

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