We received a solution from Andisheh from Spring Field School. His reasoning was correct but he used the wrong figures, so we are using his response as a basis for the solution:

Imagine extending the radius so that you have a horizontal diameter.

The hexagon is now split into two identical trapeziums (trapezia?).

Area of one trapezium

=  1 2 height x sum of parallel sides

=  1 2  × Ö0.75 × (2 + 1)

=  3 4 x Ö3

The area of the hexagon is therefore:  3 2  × Ö3

The area of the triangle is half the area of the hexagon.

The area of the triangle is therefore  3 4  × Ö3