Asymmetrical windmill

Assuming that all triangles are equilateral, we can get the remaining triangle side length, which I will call $x$ . By Power of a Point:

$\begin{aligned} (5)(x) &= (4)(10)\\ x &= 8 \end{aligned}$

Also note that because all the triangles are equilateral (and hence, similar), the ratio of their areas is based on the squares of their side lengths. Thus:

$\begin{aligned} \dfrac{\text{teal area}}{\text{orange area}} &= \dfrac{10^2 + 4^2}{8^2+5^2}\\ &= \dfrac{116}{89}\\ &\approx \boxed{1.3033707.} \end{aligned}$