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Pythagorean Theorem Calculator

Created by Piotr Małek and Mateusz Mucha
Reviewed by Hanna Pamuła, PhD and Jack Bowater
Last updated: Jan 30, 2024


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?"

How to use the Pythagorean theorem calculator

The Pythagorean theorem calculator is one of the most accessible tools you will come across, despite the name being scary. All you need is any two of the three sides of a right triangle, and you are all set.

Let's take a look at the steps to use our Pythagorean theorem calculator.

  1. Input leg a of the right triangle.

  2. Next, input leg b of the right triangle.

  3. And that is it. The tool will display the following results:

    a. The hypotenuse (c) of the right triangle.

    b. The area of the triangle.

    c. And the perimeter of the triangle.

  4. You don't necessarily have to input legs a and b. If you know the hypotenuse value and need to know any other legs, input the information accordingly.

  5. The default unit is centimeter (cm) for length and cm² for area. You can change these to one of the listed of options according to your requirements. Remember to change the unit before you input the values.

How about we try to understand an example? For instance, you are climbing a ladder to your roof, and you get curious about the length of the ladder. You notice that the distance from the roof to the ground is leg a, and the distance from the wall to the ladder's base is leg b. And the ladder itself is the hypotenuse. You input the measurements in the tool as leg a and b, 4 m and 3 m, respectively. (Remember to change the units from centimeters to meters first). The calculator instantly determines the hypotenuse as 5 m, the area as 6 m², and the perimeter as 12 m.

Next time you get curious about any setup that looks like a right triangle, you can use our Pythagorean theorem calculator to determine the lengths of the legs.

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 legs of a right triangle equals the square of the hypotenuse. You can also think of this theorem as the hypotenuse formula. If the legs 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 = 80.
  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 a reformulation of the Pythagorean theorem and is often 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)

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 lengths are 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.

The Pythagorean theorem calculator in the making

Indeed, all maths enthusiasts would be happy to have access to a Pythagorean theorem calculator. Even the students who have to complete their assignments would be thrilled. Now imagine how happy Mateusz and Piotr were when they decided to make a tool for one of the most sought-after mathematics concepts and successfully did so.

Mateusz Mucha is the brain behind Omni Calculator. His deep love for numbers with strategic vision and operational expertise is a testament to his career. He believes in a hands-on approach in all aspects of life, whether it is leadership, building some innovative calculator or digital product, or participating in a cycling marathon. He is a well-balanced blend of exemplary leadership and vision, with strategic thinking, innovation, and attention to detail being a few of the skills in his arsenal.

Piotr Małek is creative, athletic, and curious by nature, which makes him a person of remarkable discipline, high credibility, and determination. As a content writer, his ability to explain complex topics with a fairytale-like ease is genuinely fascinating. The diverse experiences he has gained over the years, his intellectual curiosity, and his lifestyle of continuous learning make him a credible voice in any field he is interested in exploring.

We ensure our tools are based on authentic information and proper research. After an expert makes the calculator, another expert in the field reviews it thoroughly. Then, a native language speaker proofreads the content, ensuring further refinement; only then is the tool released for our users. To learn more about our commitment to quality, please refer to our Editorial Policies page.

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.

What is the leg in an isosceles triangle with hypotenuse 10?

Each leg has length 10/√2 ≈ 7.07. To arrive at this answer, we apply the Pythagorean theorem:

  1. Recall the formula a² + b² = c², where a, and b are the legs and c is the hypotenuse.
  2. In our case, a = b, so the formula reads 2a² = c².
  3. Solving for a, we get a = c/√2.
  4. Plugging in c = 10, we get the final answer: a = 10/√2 ≈ 7.07.
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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