# How to calculate the length of the hypotenuse

Hypotenuse - mathematical term, which occurs when considering the right-angled triangles. It is the largest of its side opposite the right angle. Calculate the length of the hypotenuse of a variety of ways, including the Pythagorean theorem.

## Instruction how to calculate the length of the hypotenuse

Step 1:

The triangle is the simplest geometric closed figure consisting of three vertices, edges and sides, each of which has its own name. Hypotenuse and two leg - side of a right triangle, the length of which are connected with each other and with other values different formulas.

Step 2:

Most often, in order to calculate the length of the hypotenuse, the problem reduces to the use of the Pythagorean theorem, which reads as follows: the square of the hypotenuse equals the sum of the squares of the legs. Consequently, its length is the square root of this sum.

Step 3:

If you know only one leg, and the value of one of the two corners that are not straight, you can use the trigonometric formulas. Suppose given a triangle ABC in which AC = c - the hypotenuse, AB = a and BC = b - legs of, α - the angle between a and c, β - the angle between b and c. Then: c = a / cosα = a / sinβ = b / cosβ = b / sinα.

Step 4:

Solve the problem: find the length of the hypotenuse, if we know that AB = 3 and the angle BAC at this side is 30 ° .ReshenieIspolzuyte trigonometric formula: AC = AB / cos30 ° = 3 • 2 / √3 = √3 • 2.

Step 5:

It was a simple example of finding the longest side of a right triangle. Solve the following: the length of the hypotenuse, if the height of the BH, drawn to her from the opposite vertex is 4. It is also known that the height of the side divides into segments AH and the HC, and AH = 3.

Step 6:

ReshenieOboznachte unknown part of the hypotenuse HC = x. Once you find x, we can compute the length of the hypotenuse. Thus, AC = x + 3.

Step 7:

Consider the triangle AHB - it is, by definition, a rectangular height. Do you know the lengths of two legs of it, then you can find the hypotenuse a, which is the leg of the triangle ABC: a = √ (AH² + BH²) = √ (16 + 9) = 5.

Step 8:

Switch to the other right triangle BHC and find its hypotenuse is equal to b, that is the second to a leg of the triangle ABC: b² = 16 + x².

Step 9:

Go back to the triangle ABC and note the Pythagorean formula, write the equation for x: (x + 3) ² = 25 + (16 + x²) x² + 6 • x + 9 = 41 + x² → 6 • x = 32 → x = 16 / 3.

Step 10:

Substitute x and find the hypotenuse: AC = 16/3 + 25/3 = 3.