Notes
arcs in a circle solution

Solution to the Arcs in a Circle Puzzle

Solution by Sectors of a Circle and Lengths in an Equilateral Triangle

With the points labelled as above, consider the region bounded by the line segments $A B$, $B C$, and the arc $C A$. As the points are equally spaced around the circle, triangle $A B C$ is an equilateral triangle so angle $C \hat{B} A$ is $60^\circ$ and so the region is a sector of a circle with radius $A B$ and central angle $60^\circ$. Its area is therefore one sixth of that of a circle with radius $A B$.

Now consider the shaded region bounded by the arcs $O C$, $C A$, and $A O$. This is part of the sector considered above. To determine the area of the difference, consider the region bounded by the line segments $O B$, $B C$, and the arc $C O$ (curved to the left of the diagram). Since triangle $O D B$ is congruent to triangle $E D C$, this region has the same area as that bounded by line segments $O E$ and $E C$ and arc $C O$. This is a sector of a circle with radius $O E$ and central angle $60^\circ$.

From lengths in an equilateral triangle, triangle $A B C$ has area three times of that of triangle $O E C$. Therefore, the circle of radius $A B$ has area three times that of the circle of radius $O E$. Let us write $a$ for the area of the outer circle. Then the shaded area bounded by the arcs $O C$, $C A$, and $A O$ has area:

As the full shaded area comprises three such regions, its area is $\frac{1}{2} a$. The shaded area is thus half of the area of the full circle.