# How to calculate Proper Distance as an arc length in Schwarzschild metric?

I am trying to determine the method to calculate proper distance with constant time and radius in Schwarzschild Geometry. With only $$\theta$$ and $$\phi$$ being variable. I think it involves integrating the line element (metric) using two variables but an example of this is hard to find. If I can find the integral; I can probably look up a solution in a table or use Maple?

For a fixed $$r$$ and $$t$$, $$ds^2 = r^2(d\theta^2 +\sin^2\theta\ d\phi^2)\ ,$$ which is the same arc length as in Euclidean space. Then if $$\phi = f(\theta)$$ then $$s =r \int \left(1 + \sin^2\theta\ \left(\frac{df}{d\theta}\right)^2\right)^{1/2}\ d\theta\ .$$