A problem by minimario minimario

The following picture shows a portion of the New Year's Day decorations on the street, as the communities welcome $2014$ . The decorations comprise of an equilateral triangle, colored yellow, and a square, colored green, which share vertex B. When measured, the distance from vertex $E$ to the ground $(EH)$ is $8$ miles, while the distance from vertex $C$ to the ground $(CG)$ is $2+\sqrt{3}$ miles. Additionally, the flags are joined such that $\angle FBC=120^\circ$ .

Let the sum of the three areas (not decorated by flags) $ACG, BAD,$ and $HED$ be $k$ $\text{mi}^2$ . Find $\lceil k \rceil$ .

Note: $\lceil x \rceil$ denotes the least integer greater than or equal to $x$ .