# Arcs in a Semi-Circle +-- {.image} [[ArcsinaSemiCircle.png:pic]] > The two red arcs are the same length. What fraction of the semicircle is shaded? =-- ## Solution by [[Angle at the Centre is Twice the Angle at the Circumference]] and [[Area of a Triangle]] +-- {.image} [[ArcsinaSemiCircleLabelled.png:pic]] =-- 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.