Explain solid angle $\Omega=\int\int_S \frac{\bar r \cdot \hat n dS}{r^3}=\int\int_S \sin(\theta) d\theta d\varphi$

Question

I am trying to understand this formula from Wikipedia here. It is a generalization of radian. I am trying to do unit check but I cannot see how the units match. Steradion must be dimensioless unit (derived SI unit).

This $\int\int_S \frac{\bar r \cdot \hat n dS}{r^3}$ has area times a small area in the numerator and volume in denomirator so integral of one divided by meter. I am doing somewhere an idea-mistake. Where?

Formula from Wikipedia about solid angle

Answer 1

Think of solid angle as the fraction of a total unit sphere surface area ($4 \pi$) that a cap of the sphere is. The analogy is a fraction of the circumference of a circle that an arc of that circle is, is a circular angle.

Also note that $dS = r^2 \sin{\theta} d \theta d\phi$, and that $\hat{r} . \hat{n} = 1$ for the sphere.