Problem

The diagram shows a sector $ABC$ of a circle with centre $A$ and radius $r$ cm. The angle $BAC$ is $\alpha$ radians, where $0 < \alpha < \frac{1}{2}\pi$.

(a) It is given that the area of the triangle $ABC$ is $4\text{ cm}^2$ and the area of the sector $ABC$ is $8\alpha\text{ cm}^2$.

Find the exact area of the shaded segment.

(b) It is given instead that the length of the chord $BC$ is $\frac{1}{\sqrt{2}}r$ cm but the area of the triangle $ABC$ is still $4\text{ cm}^2$.

Find the area of the shaded segment. Give your answer correct to 3 significant figures.