The area of the 150° sector is
.. As = (1/2)*r^2*(5π/6)
The area of the white triangle in that sector is
.. At = (1/2)*r^2*sin(5π/6)
The orange shaded area is the difference between these.
.. A = As -At = (1/2)*r^2*(5π/6 -sin(5π/6))
.. = (1/2)*(27.8 in)^2*(5π/6 -1/2)
.. A ≈ 818.4 in^2