Take the right-way-up large equilateral triangle. If we work out what the area of it is we can just subtract 3, 12 and 27 to find the hexagon area.
The relationship between area A and side length s of an equilateral triangle is A = sqrt(3)/4 * s^2
The reverse of this is that s = sqrt(4A/sqrt(3))
So the side length of the 12, 48 and 27 triangles are respectively approximately 5.264, 10.529 and 7.896, summing to about 23.689.
Then the area of the right-way-up large triangle comes out as exactly 243.
The hexagon area is therefore exactly 201