jmc

Quadrilateral $CDEF$ is a rectangle with area $CD\cdot DE=20\cdot 10=200$. Triangle $ABG$ is right-isosceles with hypotenuse $BG=5+10+5=20$. Hence, $AB=BG/\sqrt{2}=20/\sqrt{2}=10\sqrt{2}$, so triangle $ABG$ has area $A{B}^2/2=(10\sqrt{2}{)}^2/2=100$. Therefore, the area of polygon $ABCDEFG$ is $200+100=\boxed{300}$.