Laura,

This isn't very straightforward if you want a purely numerical algorithm (something a computer can do unattended) though it's by no means impossible.

If you can plot the polygons and figure out which edges intersect and where, you can find the intersection as a polygon. Then IF all the vertices are lattice points (points with integer coordinates) you can use Pick's theorem

(area = number of lattice points inside the polygon plus half the number on the boundary minus 1)

Otherwise you can divide it up into triangles and add the areas of the triangles (which can be computed in various ways)

Good Hunting!
RD