A triangle has two corners with angles of # ( pi ) / 3 # and # ( pi )/ 6 #. If one side of the triangle has a length of #12 #, what is the largest possible area of the triangle?

1 Answer
Dec 7, 2017

#72sqrt3#square units

Explanation:

First, we know that the angels of a triangle has to add up to #pi#. Therefore, we know that the leftover angle is #pi-(pi/3+pi/6)=pi/2# which is a right angle. Therefore, we know that the triangle is a right triangle. We also know that this is the special "30-60-90" triangle. Therefore, we can figure out the sides while assuming 12 being one of each: the shortest, the middle, the longest.

Remember that in a "30-60-90" triangle, the shortest side is #a#, the second longest (middle) side is #asqrt3# and the longest side is #2a#.
Also, you multiply the two legs and then divide it by two to find the area of a right triangle.

When 12 is the hypotenuse, we know that the shortest side is 6 and the middle side is #6sqrt3# with the area of #18sqrt3#

When 12 is the shortest side, we don't really care about the hypotenuse. The middle side is #12sqrt3# with the area of #72sqrt3#.

When 12 is the middle side, we again don't care about the hypotenuse. The shortest side would be #12/sqrt3# which really is #4sqrt3# with the area of #24sqrt3#. Out of these three, we see that #72sqrt3# is the largest possible area.