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:
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:
c2 = a2 + b2
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 = c2
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 = c2 + 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 itso we generate the following equality:
(a + b)(a + b) = c2 + 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:
a2 + 2ab + b2 = c2 + 2ab
We simplify and finally we have that:
a2 + b2 = c2
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 = √(a2 + b2)
Now we substitute the lengths of the sides of the right triangle into the formula for calculating the hypotenuse:
c=√(32 + 42) = √(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:
a2 + b2 = c2
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 ExcelBelow 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:
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 worksso 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.
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