Okay Mur I found a few keys points which intrigued me not to answer but elaborate in a sense your question. While you ask if the earth is flat or round I have come up with the answer that is is neither "technically neither." Rather the earth is known as an Oblate Spheroid which by definition is "A body that is shaped like a sphere but is not perfectly round, especially an ellipsoid that is generated by revolving an ellipse around one of its axes." So what I am trying to say is give you yet an alternative to what you view as the only 2 possible forms flat or round.
the following is the Cartesian equation which is used to calculate an Oblate spheroid's dimensions.
For a spheroid with z-axis as the symmetry axis, the Cartesian equation is
(1)
The ellipticity of an oblate spheroid is defined by
(2)
The surface area of an oblate spheroid can be computed as a surface of revolution about the z-axis,
(3)
with radius as a function of given by
(4)
Therefore
(5)
(6)
(7)
(8)
where the last step makes use of the logarithm identity
(9)
valid for . Re-expressing in terms of the ellipticity then gives
(10)
yielding the particular simple form
(11)
(Beyer 1987, p. 131). Another equivalent form is given by
(12)
The surface area can also be computed directly from the coefficients of the first fundamental form as
(13)
(14)
Note that this is the conventional form in which the surface area of an oblate spheroid is written, although it is formally equivalent to the conventional form for the prolate spheroid via the identity
(15)
where is defined by
(16)
I'll have more ideas tomorrow