What is the relationship between the hypotenuse and leg of a 30-60-90 triangle

It turns out that in a 30-60-90 triangle, you can find the measure of any of the three sides, simply by knowing the measure of at least one side in the triangle. The hypotenuse is equal to twice the length of the shorter leg, which is the side across from the 30 degree angle.

Step-by-step explanation:

Notice that there are actually two legs, not just one, in a 30-60-90 triangle.

Since sin theta = opposite side / hypotenuse, where theta is either the 30 degree angle or the 60 degree angle.

Thus, the length of the side (leg) opposite the angle theta is

length = (hypotenuse) * (sin theta).

Another approach would be to recognize and remember that the three sides of a right triangle are multiples of 1, √3 and 2. The shortest side corresponds to 1; the longer side corresponds to √3, and the hypotenuse corresponds to 2. The shortest side is opposite the 30 degree angle; the side of middle length is opposite the 60 degree angle, and the hypotenuse is opposite the 90 degree angle. These statements are always true for a 30-60-90 triangle.