Question

A triangle has two corners with angles of `( pi ) / 3` and `( pi )/ 6`. If one side of the triangle has a length of `8`, what is the largest possible area of the triangle?

Answer 1

Area of the triangle `Delta ABC = (1/2) * 8 * 13.8564 = color(blue)(55.4256)` sq. units

Explanation:

Three angles of the triangle are `alpha = pi/3, beta = pi/6, gamma = pi - (pi/3 + pi/6) = pi/2`

It’s a right triangle with sides in the ratio `1 : sqrt3 :2`, length ‘2’ being the hypotenuse.

To get the maximum area, length ‘8’ should correspond to the smallest angle `pi/6`.

Hence the sides are `8, 8sqrt3, 8*2`

I.e. `8, 13.8564, 16`

Area of the triangle `Delta ABC = (1/2) * 8 * 13.8564 = color(blue)(55.4256)` sq. units