This calculator exists to compute the area of a triangle, where the three angles of the triangle measure 30, 60, and 90 degrees respectively, and only one other measurement is known. It works with a measure of any one side, or by inputting the area or perimeter.

Since our angles are rigorous, there is exactly one type of triangle we could give for any of these starting values.

The 30-60-90 triangle is also a right triangle.

The Formulas of the 30-60-90:

Given that X is the shortest side measure, we know we can measure out at the baseline for length X, turn an angle of 60 degrees, and have a new line that eventually intersects the line from the larger side at exactly 30 degrees. The formulas for finding the rest of the triangle from just X are the following, where Y = the long side, Z = the hypotenuse, A = area, and P = perimeter:

•	Y = X*sqrt(3)
•	Z = X*2
•	A = X^2*sqrt(3/2)
•	P = X*(3 + sqrt(3))

You might note that the 30-60-90 triangle is exactly one half of an equilateral triangle - a triangle with equal sides and 60 degree angles, where one side is X2, being the shortest side of our 30-60-90 triangle. And then you must know that the area of an equilateral triangle is equal to the area of a rectangle measuring X*Y.

It is also possible to use trigonometric functions to find the rest of the triangle using any metric, given its rigid angle construction. The 30-60-90 triangle is the special case of a triangle with its ratios all in perfect proportion to each other. It comes up surprisingly often in various contexts, from engineering to wave functions.

Nature is Made of Triangles

The humble triangle is the geometric shape without which most of our engineering would not be possible.
The ancient Greek mathematician and philosopher Pythagoras (500 B.C.) solved plenty of geometric conundrums including the theorem that now bears his name, which states that the hypotenuse (the diagonal line) of a right triangle is a square root which is equal to the squares of the other two sides.

For triangle (a,b,c) where the corner of (a|b) is 90 degrees, the measurements will satisfy the formula a2 + b2 = c2.

The triangle is the most stable form in construction. A square or shape with more sides can turn and collapse at the corners, but a triangle is stable enough that you can build almost anything out of it.

Geodesic domes, invented by visionary 20th-century architect Buckminster Fuller, are dome structures created out of triangle shapes, which have been demonstrated to be the strongest structure you can make that is held in place using weight, since the triangles distribute the load-bearing stress equally over the entire structure no matter where you press on it.