Question

A triangle has two corners with angles of `( pi ) / 3` and `( pi )/ 6`. If one side of the triangle has a length of `18`, what is the largest possible area of the triangle?

Answer 1

Largest possible area of triangle is `280.584`

Explanation:

The third angle of the triangle is `pi-pi/3-pi/6=(6pi)/6-(2pi)/6-pi/6=(3pi)/6=pi/2`.

Such a triangle will have the largest possible area if the side with length of `18` is opposite smallest angle `pi/6`. Let the other two sides be `b` and `c`. Then according to sine formula, we have

`18/sin(pi/6)=b/sin(pi/3)=c/sin(pi/2)` or

`18/(1/2)=b/(sqrt3/2)=c/1` or `36=b/(sqrt3/2)=c` and hence

`c=36` and `b=36xxsqrt3/2=18sqrt3`

As it is a right angled triangle and side opposite right angle is `c`,

area of triangle is given by `18xx18sqrt3)/2=162xxsqrt3`

= `162xx1.732=280.584`