The Earth is FAT!

You read me. If we assume it to be flat, then it must also be fat.

Indeed, a large homogenious disk (large meaning with a radius much greater than its height), according to Newton's Law of Universal Gravitation, will create a homogenous gravitational field perpendicular to the bases of the disk. The magnitude of this field, far from the edges can be easily calculated by an integral theorem that is analogous to the Gauss' Law in electrostatics (due to the same inverse square dependance on distance). Namely, if g is the gravitational field strength (vector), then the total flux of this field through any closed Gaussian surface S is proportional to the total mass M enclosed in that volume element:


Now, using a cylinder with a small basis ΔA and height just barely grater than the thickness H of the disk, we can evaluate the value of the integral on the left to be:


The total mass enclosed with this element is ρΔAH, so we can write:


On the other hand, the gravitational field strength of a homogenous spherical body with radius R at its surface is:


and the mass of the spherical body can be calculated using the density of the body:


Using the last two formulas, we can finally express the gravitational field of the spherical body at its surface as:


Now, equating the two expressions for the gravitational field strength in the two different models, and assuming the density is the same, we get:


or a third of the diameter. This hardly means a thin, but a rather thick disk, so the edge effects and the fringing field would be rather significant in this case.

As an end note, if we assume that the surface area of one of the bases of the disk is equal to the total surface area of the spherical body (4πR²), then the total mass of this disk is going to be:


i.e. this disk has twice the mass of the assumed spherical body.

No, it has infinitely more mass.