Question

A triangle has two corners with angles of `pi / 4` and `pi / 3`. If one side of the triangle has a length of `3`, what is the largest possible area of the triangle?

Answer 1

Largest possible area is `5.3233`

Explanation:

As two angles are `pi/4` and `pi/3`, third angle is `pi-pi/3-pi/4=(12pi-4pi-3pi)/12=(5pi)/12`.

For largest area, side of length `3`, say `a`, has to be opposite smallest angle which is `pi/4` and then using sine formula other two sides will be

`3/(sin(pi/4))=b/sin(pi/3)=c/(sin((5pi)/12))`

Hence `b=(3xxsin(pi/3))/(sin(pi/4))=(3xx0.866)/0.7071=3.674`

and `c=(3xxsin((5pi)/12))/(sin(pi/4))=(3xx0.9659)/0.7071=4.0980`

As area of the triangle is given by the formula

`Delta=sqrt(s(s-a)(s-b)(s-c))`, where `s=1/2(a+b+c)`

Here `s=1/2(3+3.674+4.0980)=5.386`

and area is `sqrt((5.386xx(5.386-3)xx(5.386-3.674)xx(5.386-4.098)`

= `sqrt(5.386xx2.386xx1.712xx1.288)=sqrt28.3372=5.3233`

Largest possible area is `5.3233`.