Area of Sector/Proof 2

Proof
From Area of Circle, the area of $\mathcal C$ is $\pi r^2$.

From Full Angle measures $2 \pi$ Radians, the angle within $\mathcal C$ is $2 \pi$.

The fraction of the area of $\mathcal C$ within the sector $BAC$ is therefore $\pi r^2 \times \dfrac \theta {2 \pi}$.

Hence the result.