These calculations can be carried out in the Python interpreter.
>>> import math
>>> a = 6378137.0 # semi-major axis, m
>>> c = 6356752.314245 # semi-minor axis, m
>>> e2 = 1 - (c/a)**2
>>> e = math.sqrt(e2)
>>> A_WGS84 = 2*math.pi*a**2*(1 + (1-e2)/e * math.atanh(e))
>>> A_WGS84
510065621724078.94
>>> r = 6371000. # mean radius, m
>>> A_sphere = 4 * math.pi * r**2
>>> A_sphere
510064471909788.25
>>> (A_WGS84 - A_sphere)/A_WGS84 * 100
0.00022542477707103626
That is, a difference of about 0.00023%.