Area is not shape nor does it define shape. It is just area.
All the physics and detail goes into Cl. Ugly physics is always in the dimentionless factors of any equation, All the factors with dimentions are almost always trivial - this is called dimention analysis. The the area is just a scale factor. It need not even be a real area of the wing, just a square of some well defined length of some part of the wing. Once you find out (no matter how) CL for specific shape of wing, it will be the same CL if you scale the wing size by factor of 2. The lift produced will be scaled by factor of 4 however through this length scale squared (dimention = area). You may choose to define it as the span, the width at half the span or the thickness.
Look at this simple example. We want lift which is a force, dimention = mass*length/time^2.
We know it should depend on the r = mass density of the air [mass/length^3], and on the V = velocity [length/time] and maybe another mysterious parameter...
To be able to produce something with force dimention we have to square the velocity to get the time^2 dimention, we have to multiply by density to get a mass dimention and the extra parameter has to be of length^2 dimention - lets call it S. So from dimentions alone I can guess that lift has to look like:
L = r * V^2 * S *(dimentionless coefficients)
Note that S does not have to be the real area of the wing, just a squared length scale. All the actual details of the exact shape of the wing and aerodynamics will go into dimentionless factors. Only thing to keep in mind when scaling things is that there might be other dimentionless numbers that need to be conserved (like the Rynolds number).
Bozon