Welcome to the Pythagorean theorem calculator. This Pythagorean calculator finds the missing side of a triangle as long as you're dealing with a right triangle.

Additionally, with this Pythagorean theorem calculator, you can calculate the perimeter of any right triangle and its area. If you want to calculate the area of any triangle (and not only right triangles), use our triangle area calculator.

The Pythagorean theorem is the basis for more advanced topics:

  • For example, the relationship between trigonometric functions sine and cosine (sin²x + cos²x + 1) comes from applying the Pythagorean theorem.
  • If you're dealing with vector addition, the Pythagorean theorem will tell you the resultant vector magnitude. For example, when finding the resultant velocity or acceleration.

Pythagorean theorem formula

Pythagorean theorem states that "the square on the hypotenuse equals the sum of the other two squares."

The Pythagorean theorem equation is:

a² + b² = c²

where

  • a is a side of the right triangle;
  • b is another side of the right triangle;
  • c is the hypotenuse.

From that Pythagorean theorem equation, you solve for any of the sides:

  • c = √(a² + b²)
  • a = √(c² - b²)
  • b = √(c² - a²)
A right triangle with its three sides: a, b, c.
A right triangle with its three sides: a, b, c.

From the previous equation, we can also deduct one of the most important rules of square triangles: None of the sides (a or b) can be greater than the hypotenuse c.

Calculating the area of a right triangle

To calculate the area of a right triangle, use the following formula:

Area = a×b/2

Calculating the perimeter of a right triangle

The perimeter of a triangle is simply the sum of its three sides:

perimeter = a + b + c

How to use this Pythagorean theorem calculator

  • Fill in the lengths of any two sides (c being the hypotenuse), and this Pythagorean calculator will evaluate the third one.
  • You can also input the area and one of the triangle sides (a or b), and the Pythagorean theorem calculator will find the other missing variables.

Let's look at how to use this calculator through an example. Suppose you know a = 8, b = 16, and want to find the hypotenuse. Follow these steps:

  1. Input a = 8 cm into the first box of the calculator.
  2. Input b = 16 cm into the second box of the calculator.
  3. The answer should be 17.89 cm. That's how you use this calculator to find the missing side of a triangle.

You can verify the result using the Pythagorean theorem equation:

c=a²+b²=(8 cm)²+(16 cm)²    =64 cm²+256 cm²=320    =17.89 cm²c = \sqrt{a² + b²} = \sqrt{(8\ \text{cm})² + (16\ \text{cm})²} \\\ \\\ \ \ = \sqrt{64 \ \text{cm}²+ 256 \ \text{cm}²} = \sqrt{320} \\\ \\\ \ \ = 17.89\ \text{cm}²
Luis Hoyos
A Pythagorean triangle with all sides marked
a² + b² = c²
a
cm
b
cm
c
cm
Area
cm²
Perimeter
cm
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