The simplest way to solve this problem is to first draw a picture. Here's a photograph of the curves described in your problem:
The region bounding the finite area is a square who's sides are of length $\pi$. Calculating this area is simple. Now for the trickier bit: calculating the area beneath a sine wave. This can be solved by applying the methods of integration.
Since there are four of these sine waves, $A=\pi^2-4\int_0^\pi \sin(x)\,dx=\pi^2-8= 1.8696\ldots$