jmc

Each side of the square has length $27$. Each trisected segment therefore has length $9$. We can form the octagon by taking away four triangles, each of which has area $\frac{(9)(9)}{2}$, for a total of $(2)(9)(9)=162$. The total area of the square is $27^2=729$, so the area of the octagon is $729-162=\boxed{567}$.