composite figures area problem

Consider a region like the ones painted red, blue or green in the given diagram. Its area is the difference between the area of a rectangle and a semi-circle, and if the radius of the semicircle is $r$ , then the rectangle has dimensions $2r \times r$ . Thus the area such a region can be written as

$2r^2 - \dfrac{1}{2} \pi r^2 = \dfrac{r^2}{2} (4 - \pi)$

Re-writing the given areas in this form, we get

$\begin{array}{rlll} A_{RED} &= \ \ 12.5 - 3.125 \pi &= \ \ 3.125 (4 - \pi) & \implies & r_R = 2.5 \\ A_{BLUE} &= \ \ 4.5 - 1.125 \pi &= \ \ 1.125 (4 - \pi) & \implies & r_B = 1.5 \\ A_{GREEN} &= \ \ 8 - 2 \pi &= \ \ 2 (4 - \pi) & \implies & r_G = 2 \end{array}$

Thus the yellow triangle is $\ 3 \ \text{-} \ 4 \ \text{-} \ 5\$ and its area is $\dfrac{1}{2}(3)(4) = \boxed{6}$

Nice problem.