Regarding geographic area bounded by 3 lines

Question

I have 3 points on a map that is specified in terms of a latitude and longitude coordinate. For each of this point, I am given an azimuth (i.e. 0 = North, measured clockwise).

By drawing 3 lines from the 3 points along the azimuth, I can find an intersecting region. My problem is thus: how can I find an equation used to compute the area of the region (e.g. in terms of km²) given the locations and azimuths?

I hope the picture below helps in understanding my question:

Answer 1

If your surface is a plane, just use Heron's formula.

Let triangle $\triangle ABC$ have side lengths $a$, $b$ and $c$. Then $$\operatorname{Area}\{\triangle ABC\}=\sqrt{s(s-a)(s-b)(s-c)}$$ where $s:=\frac{a+b+c}{2}$.

If the surface is spherical, then you can try L'Huilier's formula together with $\mathrm{Area}=ER^2$ where $E$ is the excess angle of the triangle and $R$ is the radius of the sphere. Unfortunately the formula for the area in terms of the coordinates is quite complicated, but a computer should be able to compute it for you.