Example 2.3.12: Area of a circle

The disc is $\{0\le \rho \le b,0\le \varphi <2\pi ,z=0\}$, a piece of a surface $z=\text{const}$. Its area element is, from Eq. (2.3.25), $dS=\rho d\rho d\varphi$: a small patch with radial side $d\rho$ and arc side $\rho d\varphi$ (red in the figure). Then

The factor $\rho$ in the area element is essential: without it we would get $2\pi b$, which is not even an area. It reflects the fact that patches with the same $d\rho$ and $d\varphi$ are larger far from the centre than near it.