Calculating the hypotenuse of a triangle is very easy thanks to the Pythagorean Theorem. This theorem says that the hypotenuse is equal to the square root of the sum of the legs squared, therefore, knowing this data, we can find out the value of the hypotenuse.
With our hypotenuse calculator online you can directly apply the Pythagorean Theorem, but if you need to know the value of the other legs, we also have another way to apply it. calculator to solve triangles.
What is the hypotenuse of a triangle?
The hypotenuse is the longest side which has a right triangle and also corresponds to the side opposite the right angle (the 90º side).
In the image above, it would correspond to leg C of the triangle.
How do you calculate the hypotenuse of a triangle?
As we have commented at the beginning, to calculate the hypotenuse of a triangle knowing the measure of the two legs we have to use the Pythagorean Theorem.
This theorem states that "the square of the hypotenuse is equal to the sum of the square of the legs", therefore, to calculate the hypotenuse we should apply the following formula mathematics:
c = √(a2 + b2)
By example.Let's imagine that we have a triangle whose legs measure 2 cm and 5 centimeters respectively. Applying the previous expression we have that its hypotenuse measures:
c=√(22 + 52) = √(4 + 25) = √29 = 5.39 centimeters
As you can see, calculating this side of the triangle does not involve more difficulty than to clear the unknown in a proper way and solve the operation correctly. To do this, we we recommend that you follow these steps:
- Square the length of each cathetum
- Add the values obtained from the previous step
- Apply the square root to the result of step two.
If you follow this orderly methodology, there is very little chance of getting it wrong and if you have doubts about whether you have done it right, you can always use our calculator to compare the result you have obtained and that of our tool. If they match, congratulations!
Of course, you can use this formula for all triangles that are right angles, so it doesn't matter if it is isosceles, equilateral or any other as long as there is an angle of 90 degrees.
Calculate legs knowing the hypotenuse
If we are asked to Calculate the legs of the triangle knowing the hypotenuse. and we are not given any other data, then we automatically know that it is an isosceles triangle and, therefore, it has two equal sides. This means that the formula of the Pythagorean Theorem applied to this example will be as follows:
c2 = a2 + a2 = 2a2
The value of 'c' will be the hypotenuse and the value of 'a' will be the value of the legs that we have to calculate. Now we have to clear from the above formula the value of a:
a = c/√2
That is, if we have to to find the value of the legs of a triangle from the hypotenuse we will have to divide it by the square root of two.
Calculate hypotenuse with cathetus and angle
We can also calculate the hypotenuse of a triangle if we are given one of the legs and the angle. Here is the formula for the two possible cases.
If we are given the angle and the opposite cathetus, we can calculate it with this formula:
h = opposite cathetus / sin(θ)
In the case that we are given the adjacent leg and the angle, the formula we will use will be this other one:
h = adjacent cathetus / cos(θ)
For example, let's find the hypotenuse of a triangle of which we know that it has an angle of 30º and its opposite leg measures 20 cm. To solve it, we use the first formula and solve:
h = opposite / sin(θ) = 20 cm / sin(30) = 20 cm x 0.5 = 10 cm
Easy, isn't it? If you have any doubt or problem regarding the calculation of the hypotenuse, leave us a comment and we will help you as soon as possible.
Hypotenuse of a square
There is no hypotenuse in a squareor at least that's not the right way to name it. Actually, what you are looking for is how to calculate the diagonal of a square and in the link we have just left you will find all the details to do so.