Pythagorean Theorem Calculator

Want to calculate the length of one side of a right angle triangle? Cool, try this calculator.

a c b a 2 + b 2 = c 2 a 2 + b 2 = c c 2 - b 2 = a c 2 - a 2 = b







About This Calculator

Pythagoras' theorem (also called the Pythagorean theorem) is so named because a Greek guy called Pythagoras allegedly came up with it a long time (more than 2,500 years) ago. I say allegedly because there's actually no proof that he's the one that figured it out first, but until there's evidence to the contrary he's the one that gets credit for it.


The Pythagorean Theorem

Anyway, Pythagoras' theorem states that for a right triangle (a triangle for which one corner is a right angle, which means it measures 90 degrees), the square of the hypotenuse's length (the hypotenuse is the side that is opposite the right angle, it is always the longest side in a right triangle) is equal to the sum of the squares of the lengths of the two other sides. In more plain english that means that if you square each of the two sides that are not the hypotenuse, and then add those numbers together, the number you get will be the same as the number you get when you square just the hypotenuse by itself.

The formula for Pythagoras' theorem as it's described above looks like this:

a c b a 2 + b 2 = c 2 a 2 + b 2 = c c 2 - b 2 = a c 2 - a 2 = b a 2 + b 2 = c 2

In this formula, 'a' and 'b' are the lengths of the two sides that are not the hypotenuse, and 'c' is the length of the hypotenuse. Take a look at the picture at the top of this webpage to see a right angle triangle with the sides labelled 'a,' 'b,' and 'c.'

The formula can be easily modified to allow you to solve for any one of the variables 'a,' 'b,' or 'c,' so long as you know the other two variables. Suppose you know what 'a' and 'b' are, and want to figure out what the length of the hypotenuse 'c' is? Then you would use this modified version of the formula to calculate 'c':

a c b a 2 + b 2 = c 2 a 2 + b 2 = c c 2 - b 2 = a c 2 - a 2 = b a 2 + b 2 = c 2 a 2 + b 2 = c

Now, suppose you know the hypotenuse 'c' and also side 'b,' well then you would use the following version of the formula to figure out the length of 'a':

a c b a 2 + b 2 = c 2 a 2 + b 2 = c c 2 - b 2 = a c 2 - a 2 = b a 2 + b 2 = c 2 a 2 + b 2 = c c 2 - b 2 = a c 2 - a 2 = b

Or, alternatively, if you know the hypotenuse 'c' and also side 'a,' you can use this version of the formula to calculate the length of 'b':

a c b a 2 + b 2 = c 2 a 2 + b 2 = c c 2 - b 2 = a c 2 - a 2 = b a 2 + b 2 = c 2 a 2 + b 2 = c c 2 - b 2 = a c 2 - a 2 = b

Uses for the Pythagorean Theorem

Pythagoras' theorem is useful because it allows you to calculate the length of any side of a right angle triangle so long as you know the lengths of the other two sides.

It's also useful for making sure that things are perpendicular (i.e. at a right angle) to each other. For example, imaging that you're building two walls and you want them to meet at a right angle, here's how you can check:

  1. From the corner where the two walls meet measure 3 feet along either one of the walls and mark the location.

  2. Now, on the other wall measure 4 feet from the corner and mark the location.

  3. Now, measure the distance between the two marks. If your walls are perfectly perpendicular to one another then, according to the Pythagorean theorem, the distance between the two marks should be exactly 5 feet.