Notes
arcs in a semi-circle solution

Arcs in a Semi-Circle

Solution by Angle at the Centre is Twice the Angle at the Circumference and Area of a Triangle

In the above diagram, the point labelled $O$ is the centre of the semi-circle.

Since the angle at the centre is twice the angle at the circumference, angle $B \hat{O} C$ is $60^\circ$. The red arcs have the same length, so angle $D \hat{O} E$ is also $60^\circ$. The region between the arc $E D$ and the chord $E D$ is congruent to that defined by $C$ and $D$. The triangles $A O C$ and $O B C$ both have the same base, as it is a radius of the circle, and height, as their height is the height of $C$ above the base, so have the same area. Therefore, the combined area of triangle $A O C$ with the region defined by $E$ and $D$ is the same as the sector $B O C$. So the unshaded region has the same area as two sectors with central angle $60^\circ$. Therefore the shaded region has the same area as one such sector and so consists of $\frac{1}{3}$rd of the semi-circle.