P2.2.4Estimating the surface area of the Earth
The World Geodetic System is a set of international standards for describing the shape of the Earth. In the latest, WGS-84 revision, the Earth's geoid is approximated to a reference ellipsoid which takes the form of an oblate spheroid with semi-major and semi-minor axes \(a=6378137.0\;\mathrm{m}\) and \(c = 6356752.314245\;\mathrm{m}\) respectivelty.
Use the formula for the surface area of an oblate spheroid,
\[S_\mathrm{obl} = 2\pi a^2 \left( 1 + \frac{1-e^2}{e}\mathrm{atanh}(e) \right), \;\mathrm{where}\; e^2 = 1 - \frac{c^2}{a^2},\]
to calculate the surface area of this reference ellipsoid and compare it with the surface area of the Earth assumed to be a sphere with radius 6371 km.
Show solution
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%.