How to find the hypotenuse of a right triangle

Among the numerous calculations performed to calculate various quantities of different geometric figures, is finding the hypotenuse of the triangle. Recall that a triangle is a polyhedron with three angles. Below you will find several ways to calculate the hypotenuse of different triangles.

Initially, let's see how to find the hypotenuse of a right-angled triangle. For those who have forgotten, a triangle is called rectangular, having an angle of 90 degrees. The side of the triangle, located on the opposite side of the right angle, is called the hypotenuse. In addition, it is the longest side of the triangle. Depending on the known values, the length of the hypotenuse is calculated as follows:

• The lengths of the legs are known. Hypotenuse in this case is calculated using the theorem of Pythagoras, which reads as follows: the square of the hypotenuse is equal to the sum of the squares of the legs. If we consider a right triangle BKF, where BK and KF are legs, and FB is a hypotenuse, then FB2 = BK2 + KF2. From the foregoing it follows that in calculating the length of the hypotenuse it is necessary to erect each of the sizes of the legs in turn. Then add the digits digested and extract the square root of the result.

Consider an example: A triangle with a right angle is given. One cathet is 3 cm, the other is 4 cm. Find the hypotenuse. The solution is as follows.

FB2 = BK2 + KF2 = (3cm) 2+ (4cm) 2 = 9cm2 + 16cm2 = 25cm2. Extract the square root and get FB = 5cm.

• Known is the cathette (BK) and the angle adjacent to it, which is formed by the hypotenuse and this leg. How to find the hypotenuse of a triangle? We denote the known angle α. According to the property of a right triangle, which says that the ratio of the length of the leg to the length of the hypotenuse is equal to the cosine of the angle between this leg and the hypotenuse. Considering a triangle, this can be written as: FB = BK * cos (α).
• Known is the cathet (KF) and the same angle α, only now it will be opposite. How to find the hypotenuse in this case? We turn all to the same properties of a right triangle and find out that the ratio of the length of the leg to the length of the hypotenuse is equal to the sine of the angle opposite the leg. That is, FB = KF * sin (α).

Consider the example. The same rectangular triangle BKF with hypotenuse FB is given. Suppose that the angle F is 30 degrees, the second angle B corresponds to 60 degrees. Still known is the BK cathet, whose length is 8 cm. You can calculate the required value as follows:

FB = BK / cos60 = 8 cm.
FB = BK / sin30 = 8 cm.

• There is a known radius of a circle (R), described near a right angle triangle. How to find a hypotenuse when considering such a problem? From the property of a circle described around a triangle with a right angle it is known that the center of such a circle coincides with the point of the hypotenuse that divides it in half. In simple words, the radius corresponds to half the hypotenuse. Hence the hypotenuse is equal to two radii. FB = 2 * R. If an analogous problem is given in which the median is known not, but the median, then one should pay attention to the property of a circle circumscribed around a triangle with a right angle, which says that the radius is equal to the median drawn to the hypotenuse. Using all these properties, the problem is solved in the same way.

If there is a question, how to find the hypotenuse of an isosceles right-angled triangle, then all must be turned to the same theorem of Pythagoras. But, first of all, let us recall that an isosceles triangle is a triangle having two identical sides. In the case of a right triangle, the same sides are the legs. We have FB2 = BK2 + KF2, but since BK = KF we have the following: FB2 = 2 BK2, FB = BK√2

As you can see, knowing the Pythagorean theorem and the properties of a right triangle, it is very simple to solve the problems in which it is necessary to calculate the length of the hypotenuse. If all the properties are difficult to remember, learn the ready formulas, substituting into which the known values can be calculated the desired length of the hypotenuse.