So, this is a question of simple geometry, if I understand it right :-)

If you know the A and C vectors in terms of xyz (as shown in the Lattice Parameters plugin) all you need is the area formula area = (ac/2)*sin(beta) where a and c are the lengths of the A and C lattice vectors, and beta the angle between them.

For an example, take the Chromium Bromide (5/3) crystal you can find in the Database. It's a body-centered tetragonal lattice, so when you first open it, it shows lattice constants a=b=5.46 Å, c=10.64 Å and beta = 90, but this doesn't quite help us, as this actually refers to the conventional cell.

Therefore, convert it to a UnitCell, which will show the lattice parameters of the primitive cell: a=b=c=6.57 Å and beta=130.9 degrees. So the area of the AC plane is 6.57^2/2*sin(130.9) or about 16.3 Å^2 (if I got the numbers right; I will leave that as an exercise to verify ;-) ).