Pythagorean Theorem Calculator

Created by Piotr Małek and Mateusz Mucha
Reviewed by Hanna Pamuła, PhD and Jack Bowater
Last updated: Jan 14, 2023

This Pythagorean theorem calculator will calculate the length of any of the missing sides of a right triangle, provided you know the lengths of its other two sides. This includes calculating the hypotenuse. The hypotenuse of the right triangle is the side opposite the right angle, and is the longest side. This side can be found using the hypotenuse formula, another term for the Pythagorean theorem when it's solving for the hypotenuse. Recall that a right triangle is a triangle with an angle measuring 90 degrees. The other two angles must also total 90 degrees, as the sum of the measures of the angles of any triangle is 180. Read on to answer "what is the Pythagorean theorem and how is it used?"

What is the Pythagorean theorem?

The Pythagorean theorem describes how the three sides of a right triangle are related in Euclidean geometry. It states that the sum of the squares of the sides of a right triangle equals the square of the hypotenuse. You can also think of this theorem as the hypotenuse formula. If the sides of a right triangle are a and b and the hypotenuse is c, the formula is:

a² + b² = c²

The theorem was credited to the ancient Greek philosopher and mathematician Pythagoras, who lived in the sixth century BC. Although it was previously used by the Indians and Babylonians, Pythagoras (or his students) were credited with being the first to prove the theorem. It should be noted that there is no concrete evidence that Pythagoras himself worked on or proved this theorem.

How do I use Pythagorean theorem?

Here's how to use Pythagorean theorem:

  1. Input the two lengths that you have into the formula. For example, suppose you know one leg a = 4 and the hypotenuse c = 8.94. We want to find the length of the other leg b.
  2. After the values are put into the formula, we have 4²+ b² = 8.94².
  3. Square each term to get 16 + b²= 80².
  4. Combine like terms to get b² = 64.
  5. Take the square root of both sides of the equation to get b = 8. Go ahead and check it with an online Pythagorean theorem calculator!

Note that if you are solving for a or b, rearrange the equation to isolate the desired variable before combining like terms and taking the square root

The Pythagorean theorem calculator will solve for the sides in the same manner that we displayed above. We have included the method to show you how you can solve your problem if you prefer to do it by hand.

What is the hypotenuse formula?

The hypotenuse formula simply takes the Pythagorean theorem and solves for the hypotenuse, c. To solve for the hypotenuse, we simply take the square root of both sides of the equation a² + b² = c² and solve for c. When doing so, we get c = √(a² + b²). This is just an extension of the Pythagorean theorem and often is not associated with the name hypotenuse formula.

Other considerations when dealing with triangles

Notice the sides of a triangle have a certain degree of gradient or slope. We can use the slope calculator to determine the slope of each side. In a right triangle, the sides that form the right angle will have slopes whose product is -1. The formula for slope, if you wish to calculate by hand, is:

(y₂ - y₁)/(x₂ - x₁)

So if the coordinates are (3,6) and (7,10), the slope of the segment is (10-6)/(7-3) = 1. If the slope of the other segment forming the angle is -1, then the lines would be perpendicular since 1 × -1 = -1. Therefore, the triangle is a right triangle.

You can also figure out the missing side lengths and angles of a right triangle using the right triangle calculator. If the angles given in the problem are in degrees and you want to convert to radians or radians to degrees, check out our angle converter. There is an easy way to convert degrees to radians and radians to degrees.

If the angle is in radians:

  • Multiply by 180/π.

If the angle is in degrees:

  • Multiply by π/180.

Sometimes you may encounter a problem where two or all three side lengths missing. In such cases, the Pythagorean theorem calculator won't help – you will use trigonometric functions to solve for these missing pieces. Don't worry! We have an excellent trigonometric functions calculator available for you.

FAQ

What is the hypotenuse given legs 7 and 9?

The hypotenuse is 11.40.

You need to apply the Pythagorean theorem:

  1. Recall the formula a²+ b² = c², where a, and b are the legs and c is the hypotenuse.
  2. Put the length of the legs into the formula: 7²+ 9² = c².
  3. Squaring gives 49 + 81= c². That is, c² = 150.
  4. Taking the square root, we obtain c = 11.40. You can verify the result with an online Pythagorean theorem calculator.
Piotr Małek and Mateusz Mucha
A Pythagorean triangle with all sides marked
a² + b² = c²
a
cm
b
cm
c
cm
Area
cm²
Perimeter
cm
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