Thanks to the **Pythagorean theorem** we know that the value of the hypotenuse squared of any right triangle is equal to the sum of the squares of the legs. This can be seen visually and mathematically in the image above, in which you can see the legs 'a' and 'b' together with the hypotenuse 'c' which corresponds to the longest side.

If you want to calculate the value of any side of a right triangle knowing the other two, now you can do it thanks to our** Pythagorean Theorem online calculator:**

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## Formulas of the Pythagorean Theorem:

As we said at the beginning, **the Pythagorean Theorem states** that:

In any right triangle, the sum of the square of the hypotenuse is equal to the sum of the square of its legs.

This can be represented mathematically by the following **formula**:

c^{2} = a^{2} + b^{2}

From here, we can calculate the length of the hypotenuse or either of the two legs if we know the value of two sides.

**Cathetus a**:**Cathetus b**:**Hypotenuse c**:

As you can see, you only have to clear the unknown of the side of the triangle we are interested in and solve the formula that includes a square root and for which you can use our square root calculator.

## Demonstration of the Pythagorean Theorem

For **prove the Pythagorean Theorem** we are going to use the graph above these lines. In it we can see a square inside of which we have four right triangles.

The area of the square is calculated by multiplying the base by its height, which are exactly equal so we have that:

Area = (a + b)(a + b)

Now we calculate the areas of the inner square and the four right triangles:

Small square area = c

^{2}Area of a triangle: = ab/2

Area of the 4 triangles = 4 (ab/2) = 2ab

Now we add the area of the triangles and the small square:

Total area = c

^{2}+ 2ab

If we look at it, **the area of the large square is equal to the sum of the areas of all the figures inside it**so we generate the following equality:

(a + b)(a + b) = c

^{2}+ 2ab

The next step for the proof of the Pythagorean Theorem consists of **grouping all the unknowns of the same type** and for that, we have to develop:

a

^{2}+ 2ab + b^{2}= c^{2}+ 2ab

We simplify and finally we have that:

a

^{2}+ b^{2}= c^{2}

Does the formula sound familiar? Yes, it does, **the Pythagorean Theorem has been demonstrated**.

## Applications of the Pythagorean Theorem

**The Pythagorean Theorem has different applications** depending on the data we have from the triangle:

### Calculate the hypotenuse by knowing the two legs

As we have said before, if we have the length of the two legs of the triangle, we can** calculate the value of the hypotenuse** by applying the following formula:

Remember that for **calculate the hypotenuse** it is essential that the triangle be a right triangle since it is the only one that has a right angle or 90º. That is, an isosceles or equilateral triangle has no hypotenuse.

It is also common to make the mistake of wanting to** find the hypotenuse** of a rectangle but that is not correct. If we are talking about a rectangle or a square, what we really want to calculate is its diagonal.

If you already know what the hypotenuse of a right triangle is, we are now going to see an exercise in which you will see **how to calculate the hypotenuse** using the formula we have seen before. To do this, let's imagine that we have a triangle whose legs measure 3 and 4 centimeters respectively. How long will the hypotenuse measure? Let's calculate it:

c = √(a^{2} + b^{2})

Now we substitute the lengths of the sides of the right triangle into the **formula for calculating the hypotenuse**:

c=√(3^{2} + 4^{2}) = √(9 + 16) = √25 = 5 centimeters

As you can see, **calculating the hypotenuse does not entail much difficulty**. You just have to correctly square each term, perform the sum, and then add it up, make the square root of the result obtained.

### Calculate one leg knowing the other leg and the hypotenuse.

If we know the length of one of the legs and the hypotenuse of the right triangle, we can calculate the value of the missing side by applying one of the two formulas below (choose the one that applies in each case):

**Cathetus a**:**Cathetus b**:

### To know if the triangle is rectangular

So that **a triangle is right-angled** it must be satisfied that the sum of the square of the legs is equal to the square of the hypotenuse, that is:

a^{2} + b^{2} = c^{2}

Therefore, if we know the lengths of the three sides of a rectangle, we can** calculate whether it is rectangular or not** checking that the above equality is satisfied. If it is not satisfied, it is not a right triangle.

## Pythagorean Theorem Exercises

Taking into account the theory that you have seen, you can create your own** Pythagorean Theorem exercises** and check your result with our calculator.

All you have to do is **invent the length of two of the sides and apply the corresponding formula** to get the result of the missing one. It is very easy and you can do it yourself to check that you have understood well the Pythagorean Theorem.

## Pythagorean Theorem in Excel

If you want to **solving the Pythagorean Theorem using Excel**Below you will find the formulas for using the Microsoft spreadsheet program.

As these are formulas that we have used as an example, if you do not want to have problems, try to place the data in the same cells that we have used, otherwise, you will have to modify their coordinates and if you do it wrong, you will not get the correct results.

- Leg A: cell C5
- Leg B: cell C6
- Hypotenuse C: cell C7

Now are the formulas for calculating the result of each of the sides of the triangle by applying the **Pigator's Theorem in Excel**:

**A. Cateto:**

=RAIZ(C7^2-C6^2)

**Cateto B:**

=RAIZ(C7^2-C5^2)

**Hypotenuse C:**

=RAIZ(C5^2+C6^2)

With this, you can solve any right triangle using the** Pythagorean Theorem from Excel**. Enter the value of two of the three unknowns in the source cells (C5, C6 or C7) and you will automatically get the result.

## How does the Pythagorean Theorem calculator work?

Above you have a video in which we explain you in detail **how the Pythagorean Theorem online calculator works**so you won't have any doubts about it. If after watching the video you still don't know how to calculate the leg of a right triangle or its hypotenuse using our tool, leave us a comment and we will help you as soon as possible.

If the information we have provided here has helped you, we would greatly appreciate it if you would subscribe to our social networks or share this calculator among your friends :D

Excellent

Seriously, you are so bored with life.

Hello Ivan,

Not really, I am not bored at all. However, I cannot say the same about you since you have entered my web site, you have reached the end of the long post with information about the Pythagorean Theorem and you have taken the trouble to answer.

Who has time to spare? Thank you!

Thanks for the procedure to apply it in Excel ;D

Your support is very important, I need to find adjacent leg of hypotenuse 10 and opposite leg 1.

Hello Arturo,

To calculate the adjacent leg just apply this formula:

b = √(c

^{2}- a^{2}) = √(100 - 1) = √99 = 9,9498 cmThat is the result you are looking for.

If I know the value of the hypotenuse and the value of the sum of the legs, how do I do it?