Question: if there is a triangle with a 30, 60, and 90 degree angle and the shortest side is 6cm how do you find the area?

The easiest way to calculate the area of a right triangle (a triangle in which one angle is 90 degrees) is to use the formula A = 1/2 b h where b is the base (one of the short sides) and h is the height (the other short side).

You know the shortest side length but you need to find the other leg of the triangle. Since you also know the measure of all the angles, you can use the Law of Sines to find the missing length. The Law of Sines says that the ratio of the sine of an angle to the length of the side opposite that angle is a constant for all angles and sides in any
triangle. That is, if you have a triangle with sides labelled a, b, and c and the angles opposite those sides are labelled A, B and C, then:

sin(A)/a = sin(B)/b = sin(C)/c.

So put in the angles and the side you know and you will get the other short side (leg) of the right triangle. Then use that as h in the area formula along with the shortest side length you were given as b.

The Law of Sines is valid for any triangle but your triangle is quite special (on of the angles is 90 degrees) and there is another way to find the second short side. You know that the shortest side is 6 cm long so this side is opposite the smallest angle, that is the 30 degree angle.

Suppose that the second short side is a cm as in the diagram then

tan(30^o) = 6/a

Hope this helps,
Stephen and Penny