How to compute the area of a polygon
- If it's convex or star-shaped, you can pick a central point
  and chop it up into triangles (with this vertex and each side a base),
  and compute the area of each
  (if you're given points in R^2, Heron's formula is the most direct
   way to get area)

- This works even if it's not star-shaped!
  You simply -subtract- if the triangle goes the other way
  (going counterclockwise, say, if your next vertex is
   counterclockwise of the previous, you add, otherwise you subtract)
  (if you're given points in R^2, you can get this by comparing
  slopes (b_1/a_1 < b_2/a_2; more accurately, by ordering them on the
  projective line at infinity, so a_2b_1 < a_1b_2)
[yeah, I figured this out on-the-spot at my Goldman Sachs interview;
 I didn't realize that it was essentially the same as the star-shaped
 case, though I gave the right answer.]

- if it's a curve, you simply integrate
  (details: ? basically "area in polar coordinates")

Somewhat surprisingly, this allows you to define the "area" of a
-self-intersecting- curve! (Just use the same formula)
This isn't magic; it's the same as the area you'd get if you chop it
up into several overlapping discs (glue together where it crosses),
and you get what you'd expect.
Note that -in the computation-, you use signed area:
this is b/c you need to choose an orientation for the path,
and the opposite orientation would yield the opposite sign for the
area, though we can think of the area as unsigned (take absolute
value).