This pythagoren theorem calculator lets you calculate the length of the hypotenuse of a right triangle if you know the length of its legs. You may also use the calculator to calculate a leg or the area of a right triangle.

The calculator will give you not just the answer, but also a step by step solution.

## Usage Guide

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#### i. Valid Inputs

The pythagorean calculator needs exactly two out of the three inputs.

Each of the inputs provided must be a non-negative real number. In other words, the input must be 0 or greater. Here are a few examples.

- Whole numbers or decimals → $\hspace{0.2em} 2 \hspace{0.2em}$, $\hspace{0.2em} 4.25 \hspace{0.2em}$, $\hspace{0.2em} 0 \hspace{0.2em}$, $\hspace{0.2em} 0.33 \hspace{0.2em}$
- Fractions → $\hspace{0.2em} 2/3 \hspace{0.2em}$, $\hspace{0.2em} 1/5 \hspace{0.2em}$
- Mixed numbers → $\hspace{0.2em} 5 \hspace{0.4em} 1/4 \hspace{0.2em}$

#### ii. Example

If you would like to see an example of the calculator's working, just click the "example" button.

#### iii. Solutions

As mentioned earlier, the calculator won't just tell you the answer but also the steps you can follow to do the calculation yourself. The "show/hide solution" button would be available to you after the calculator has processed your input.

#### iv. Share

We would love to see you share our calculators with your family, friends, or anyone else who might find it useful.

By checking the "include calculation" checkbox, you can share your calculation as well.

Here's a quick overview of what the pythagorean theorem is and how we can use it to solve right triangles.

For those interested, we have a more comprehensive tutorial on the pythagorean theorem.

## The Pythagorean Theorem

The first thing to understand about the pythagorean theorem is that it applies only to right triangles. A right triangle is a triangle in which one angle is a right angle.

The pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs.

$c^2 = a^2 + b^2$

The theorem allows us to easily find the third side if we know any two sides of a right triangle.

Example

If the hypotenuse and one leg of a right triangle meansure $\hspace{0.2em} 10 \text{ in} \hspace{0.2em}$ and $\hspace{0.2em} 6 \text{ in} \hspace{0.2em}$ respectively, find the length of the other leg.

Solution

According to the pythagorean theorem, in a right triangle with a hypotenuse $\hspace{0.2em} c \hspace{0.2em}$ and legs $\hspace{0.2em} a \hspace{0.2em}$ and $\hspace{0.2em} b \hspace{0.2em}$ —

$c^2 = a^2 + b^2$

Substituting the values, we have

$\begin{align*} 10^2 \hspace{0.25em} &= \hspace{0.25em} 6^2 + b^2 \\[1em] b^2 \hspace{0.25em} &= \hspace{0.25em} 10^2 - 6^2 \\[1em] b^2 \hspace{0.25em} &= \hspace{0.25em} 36 \end{align*}$

Taking the square root on both sides,

$b = 6$

We ignore the negative root because the side length would be positive.