Pythagorean Theorem Calculator

- May 10, 2025
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The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs).

Formula: a² + b² = c² where c is the hypotenuse and a and b are the legs.

What would you like to calculate?

units
? Enter the length of the first leg of the right triangle.
units
? Enter the length of the second leg of the right triangle.

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About the Pythagorean Theorem

The Pythagorean theorem is one of the most fundamental and well-known theorems in mathematics. It was named after the ancient Greek mathematician Pythagoras (c. 570–495 BC), although evidence suggests that knowledge of the theorem predates him.

The Theorem:

In a right triangle, 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 (legs).

c² = a² + b²

This relationship is illustrated in the famous visual proof where squares are drawn on each side of a right triangle, showing that the area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides.

Applications:
  • Distance calculations: Finding the shortest distance between two points in a plane.
  • Construction: Ensuring right angles in building and surveying.
  • Navigation: Calculating distances and coordinating positions.
  • Physics: Vector calculations and resolving forces.
  • Engineering: Structural analysis and design.
Pythagorean Triples:

These are sets of three positive integers a, b, and c that satisfy the equation a² + b² = c². The most well-known Pythagorean triple is (3, 4, 5).

Other examples include:

  • (5, 12, 13)
  • (8, 15, 17)
  • (7, 24, 25)
Extensions and Generalisations:
  • Law of Cosines: Generalises the Pythagorean theorem to non-right triangles.
  • Higher Dimensions: In 3D space, the distance formula becomes √(x² + y² + z²).
  • Non-Euclidean Geometry: Different versions of the theorem apply in spherical and hyperbolic geometries.

Understanding the Pythagorean Theorem

The Pythagorean theorem is a simple yet essential concept in Geometry. It deals with right triangles, specifically, those that have one angle measuring 90 degrees. According to the theorem, the square of the hypotenuse's length (the side opposite the right angle) equals the sum of the squares of the Other two sides. This relationship is summarised by the equation a² + b² = c², where 'c' signifies the hypotenuse, and 'a' and 'b' are the triangle's legs.

What Is a Pythagorean Theorem Calculator?

A Pythagorean theorem calculator is a handy tool designed to help you find missing side lengths in right triangles. Whether you need to calculate the hypotenuse or one of the legs, this calculator simplifies the process. By inputting known values, you can quickly get results without having to do the math yourself.

Features of the Calculator

  • Multiple Calculation Options: Choose to calculate the hypotenuse, a missing leg, or verify if a triangle is a right triangle.
  • Visualisation: The tool can display a triangle diagram to help you understand the dimensions better.
  • Step-by-Step Results: You can opt to view the calculation steps, making it easier to follow along.

How to Use the Calculator

Using the Pythagorean theorem calculator is straightforward. First, select what you want to calculate—whether it’s the hypotenuse, one of the legs, or a verification check. Enter known values for the sides as prompted. The calculator will then provide you with the results and any additional properties of the triangle, such as area and perimeter.

Applications of the Pythagorean Theorem

The Pythagorean theorem has numerous real-world applications. Here are a few ways it’s commonly used:

  • Construction: Builders use it to ensure structures are square.
  • Navigation: It helps in determining the shortest path between two points.
  • Physics: It assists in calculating forces and motion in various scenarios.

Benefits of Using the Calculator

One of the main advantages of using the Pythagorean theorem calculator is accuracy. Manual calculations can lead to errors, especially with decimals. The calculator ensures precision and saves time. Additionally, users can learn the principles of geometry better by seeing how changes in lengths affect the triangle's properties.

Exploring Pythagorean Triples

Pythagorean triples are sets of three integers that satisfy the Pythagorean theorem. Common examples include (3, 4, 5) and (5, 12, 13). These triples are particularly useful in various mathematical problems and can also help in determining integer solutions for triangles without needing decimals.

Learn More About the Theorem

The Pythagorean theorem is more than just a mathematical formula; it's a gateway to understanding geometry. By exploring this calculator, you not only get accurate calculations but also gain insights into other related concepts, such as the law of cosines and triangle properties. Engaging with these ideas can deepen your appreciation for Mathematics.