Solid angle in the spherical polar co-ordinate system:

In spherical polar co-ordinate system, there are three area elements ${dA}_r,\ {dA}_{\theta}$ and ${dA}_{\phi}$, out of which only the area element ${dA}_r$ subtends a solid angle at the centre $O$.

[To know about the three types of area elements (CLICK HERE)]

The area element ${dA}_r$ is perpendicular to the direction of the radial vector \vec{r}, Hence only the area ${dA}_r$ subtends a solid angle at the centre O of the sphere.

[To know in detail, why only the area element ${dA}_r$ subtends a solid angle at the centre (CLICK HERE)]

We know that the value of this area element ${dA}_r$ is given by,

${dA}_r=r^2\ \sin\theta\ d\theta\ d\phi$.

So the solid angle subtended by the area ${dA}_r$ is given by,

$d\omega=\displaystyle{\frac{ r^2\ \sin\theta\ d\theta\ d\phi }{r^2}}= \sin\theta\ d\theta\ d\phi$