# 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:

1. How would one theoretically calculate the area between four points on a sphere, and could this be applied to an ellipsoidal shape?
2. Apologies if this does not fit on the Math stackexchange, but how is land area calculated in the real world?
• I think in the real world, if someone actually wanted to go out and measure a land area, they would do it in patches small enough that the curvature of the Earth didn't really matter, and then add them together. That way you break one complicated calculation into many simple calculations instead. And performing many simple calculations is exactly the kind of thing a computer excels at. – Arthur Jun 19 at 9:21
• Perhaps a good question for : gis.stackexchange.com – Floris Claassens Jun 19 at 12:42

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.