Math and Arithmetic
Geometry
Trigonometry

# How do you find the length of the hypotenuse?

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In a right-angled triangle, the hypotenuse is the longest side, opposite the right-angle. There are two ways of finding the length of the hypotenuse using mathematics: Pythagoras' theorem or trigonometry, but for both you need either two other lengths or an angle.

For Pythagoras' theorem, you need the other two lengths. The theorem is a2+b2=c2, or the square root of the sum of two angles squared, where c=the hypotenuse. Let's say that one length is 4.8cm and the other 4cm. 4.82+42=6.22. So, the answer is 6.2cm.

If you have one side and one angle, use trigonometry. You will need a calculator for this. Each side of the right-angled triangle has a name corresponding to the positioning of the angle given. The opposite is the side opposite the given angle, the adjacent is the side with the right-angle and the given angle on it, and the hypotenuse is the longest side or the side opposite the right-angle. There are three formulas in trigonometry: sin, cos and tan. Sin is the opposite/hypotenuse; cos is the adjacent/hypotenuse; and tan is the opposite/adjacent. As we are trying to find the hypotenuse, we already have either the opposite or the adjacent, and one angle. Let's say that our angle is 50o and we have the adjacent side, and that is 4cm. So, we have the adjacent and want to know the hypotenuse. The formula with both the adjacent and the hypotenuse in is cos. So, Cos(50o)=4/x where x=hypotenuse. We can single out the x by swapping it with the Cos(50o), so x=4/Cos(50o) -> x=6.22289530744164. This is the length of the hypotenuse, and is more accurate that Pythagoras' theorem.

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