Everyone knows that the circumference of Sigil is 20 miles, but that's not quite enough information to define it fully. A torus has two radii: r for the small generator circle and R for the big circle the small one revolves around. Also, Sigil is not a complete torus: the inner third of the generating circle has been cut out.
Now, the known circumference constrains r and R, so that if you know one, you can calculate the other. But what value produces a Sigil with the proper "thickness"? It should look more like an inner tube than a bicycle tire.
The best way to measure it is by looking at a flat rectangular map of Sigil. Define a as the ratio between the length of one of the rectangles and its height. (Remember, the circumference is equal to twice the length of the rectangle.) After doing a few calculations, you can find r, R, and the surface area A to two significant figures:
r = 2.4 / a (in miles)
R = 3.2 - r (in miles)
A = 200 / a - 88 / a^2 (in square miles)
Now, on the published map, it looks like a = 4, but when I run that through my computer animation program, it produces a very "thin" Sigil. It looks like a = 3 produces better results. This means that:
r = 0.80 miles
R = 2.4 miles
A = 57 square miles (a bit smaller than Liechtenstein)
There you have it!
In the Cage: A Guide to Sigil says that the City of Doors has a diameter of five miles (page 6). If that's the diameter of your small generator circle, then that circle would have a radius of 2.5 miles.
That would be an impossible shape (since a 20-mile circumference circle would have a diameter of about 6.4 miles, while a torus whose tube is five miles in diameter must be at least 10 miles across), though of course impossible shapes aren't strictly beyond the realm of possibility in Planescape. I believe that gives us an area of 67.419155331745358375402063331018 square miles, or would if Sigil's torus was complete, which it isn't, and if the shape I described was possible, which it isn't.
If Sigil isn't an actual torus, but a ring whose latitudinal curve is much less acute than a section of a circle would be, I'm pretty sure it's possible, but it means we can't use the formula to calculate the area of a torus to figure out how big it is.
Sigil's size changes continuously anyway, and, if space itself is distorted within, it might be bigger than its dimensions suggest.
I'd be very interested in seeing screenshots of Sigil's shape from your animation program (especially as seen from inside), regardless of what numbers you use.