How is land area calculated when the ellipsoidal shape of the Earth cannot be neglected? I was curious as to how the land area of a state such as Colorado could be calculated. I understand the area of a 2D rectangle can be calculated using the formula width times length. However, I was wondering how being on a curved surface affects this area.
For example, Wikipedia states the length, width, and area of Colorado to be 610km, 450km, and 269,837 square kilometers. Assuming Colorado is perfectly rectangular, the area by multiplying the length and width is 274,500 square kilometers, which is greater than the quoted area.
So, I have a few questions:


*

*How would one theoretically calculate the area between four points on a sphere, and could this be applied to an ellipsoidal shape?

*Apologies if this does not fit on the Math stackexchange, but how is land area calculated in the real world?

 A: Colorado is not perfectly rectangular, and its northern border (assuming that it runs along a parallel) is shorter than the southern one. To see this it is sufficient to consider the Earth as an ideal sphere, in which case:
$$
L_{E/W}=R|\theta_N-\theta_S|,\quad L_{S/N}=R|\phi_W-\phi_E|\cos\theta_{S/N},\quad A=R^2|(\phi_W-\phi_E)(\sin\theta_N-\sin\theta_S)|,
$$
where $R$ is the Earth radius, and $\phi_{E/W}$  and $\theta_{S/N}$ are longitudes and latitudes of the borders (the angles are given in radians). Applying this expression one obtains:
$$
\begin{array}{}
R & L_S & L_N & L_{E/W} & A\\
6378 &  622.3 & 588.1 & 445.3 & 269585\\
6356 &  620.2 & 586.1 & 443.7 & 267729
\end{array}
$$
where two values (equatorial and polar) are given for the Earth radius. $L_S,L_N,L_{E/W}$ and $A$ are the lengths (in km) of southern, northern, eastern/western borders  and the area (in km$^2$), respectively.
As seen already the lengths of borders given in wikipedia do not match. Also observe that we do not take the real relief (mountains and so on) into account, and the latter effect can be comparable in the value with that one resulting from the deviation from the spherical form.
