How to calculate specific area on surface of sphere?

I am trying to calculate the solid angle subtended by arbitrary-shaped loops on a sphere's surface.

First, I parametrize circular loops by:

$$\theta(t,k_{x0},r) = k_{x_0} + r \cos(t);$$ $$\phi(t,k_{y0},r) = k_{y_0} + r \sin(t);$$

where $$0\leq t\leq2\pi$$, and $$k_{x_0}$$, $$k_{y_0}$$ define the loop's center. So, we can say that this step draws out a circular loop on the $$\theta/\phi$$ plane.

Then I project these onto the sphere's surface using spherical coordinates, as follows:

$$x(\theta,\phi)=r \cos{\theta}\sin{\phi}, y(\theta,\phi)=r \sin{\theta}\sin{\phi}, z(\theta,\phi)=r \cos{\phi}$$

How do I go about calculating the surface area within these $$(x,y,z)$$ loops on the surface? This will allow me to calculate the solid angle I need.

The solid angle is given by: $$\Omega= \iint_S \frac{\hat{r}\cdot\hat{n}}{r^2} \, \mathrm{d}\Sigma = \iint_{\mathcal{R}}\sin \theta \, \mathrm{d} \theta \, \mathrm{d} \phi=\frac{\textrm{spherical surface area}}{r^2}$$

I tried using various types of RegionMeasures to calculate this area (such as defining the area within the loop on the sphere as a Region, and by varying the radius from 0 to r, calculating the length of each loop in between and summing it all up), but I feel like I am missing a simple answer to my problem. Maybe what I am missing is a way to somehow map my arbitrary loops into a appropriate integration bounds for $$\Omega$$, but I tried to avoid this by resorting to Mathematica.

So far, I found the following posts most useful:

Integrate to calculate enclosed area

https://math.stackexchange.com/questions/1832110/area-of-a-circle-on-sphere

Note: I am parametrizing these loops in a peculiar way because I am trying to investigate a physics problem where the functions $$x(\theta,\phi),y(\theta,\phi),z(\theta,\phi)$$ will be different, and make my loops twist and turn. The ultimate goal is to find the solid angle in these cases, but I wanted to start with the sphere.
• One function, I don't know if you've seen it, that could perhaps compute the solid angle with your parametrization is Area together with spherical coordinates, which can be specified as the third argument. – C. E. Feb 27 at 21:14