I will examine a 100-sided regular polygon. If you split it into 100 pizza slice triangles, then each one has a base of 1 foot (with central angle of 3.6 degrees), and the other two sides are equal. Split it down the middle to form two right triangles, each with base 0.5 across from an 1.8 degree angle.
To find the area, we need the height of the triangle (and since this is a right triangle, we need the other leg). Since the tangent ratio gives us opposite over adjacent, we have
tan(1.8)=0.5/x ---> x = 0.5/tan(1.8)
Now we have the height of each pizza slice, so we can find the area:
A = 1/2*1*x = x/2
And multiplying this by 100 gives us the polygon's area:
A = 50x = 795.51ft^2... pretty close to the circle!