How do you find the height of an equilateral triangle if you have the length of the hypotenuse?

Answer 1

An equilateral triangle hasn't a hypotenuse; hypotenuse means the side opposite the right angle in a right triangle. An equilateral triangle has no right angles; rather all three of its angles measure 60 degrees. Knowing the length of the hypotenuse of a right triangle does not give enough information to determine the triangle's height. But the length of a side (which is the same for every side) of an equilateral triangle is enough information from which to calculate the height of that triangle. The first way is simply to use the formula that has been developed for this purpose: height = (length X sqrt(3)) / 2. But you can also use the geometry of right triangles to solve for the height. That is because you can bisect the triangle with a vertical line from the top vertex to the center of the base. The length of that line, which splits the equilateral triangle into two right triangles, is the height of the equilateral triangle.

We know a lot about each right triangle formed by bisecting the equilateral triangle: * - The hypotenuse length is the length of the equilateral triangle's side. * - The base length is half the length of the hypotenuse. * - The angle opposite the hypotenuse is 90 degrees. * - The angle opposite the vertical is 60 degrees (the measure of every angle of any equilateral triangle). * - The angle opposite the base is 30 degrees (half of the bisected 60-degree angle). * - (Note that the sum of the angles does equal 180 degrees, as it must.) Now to solve for the height of a right triangle. There are a few ways. For labeling, let's let h=height of the equilateral triangle and the vertical side of the right triangle; A=every angle of the equilateral triangle (each 60o); s=side length of any side of the equilateral triangle and thus the hypotenuse of the right triangle.

Since the sine of an angle of a right triangle is equal to the ratio of the opposite side divided by the hypotenuse, we can write that sin(A) = h/s. Solving for h, we get h=sin(A)/s. With trig tables you can now easily find the height.