In the diagram, regular hexagon \(ABCDEF\) has sides of length \(2\).
Using \(A\), \(C\) ,\( E\) as centers, portions of circles with radius \(1\) are drawn outside the hexagon.
Using \(B\), \(D\) and \(F\) as centers, portions of circles with radius \(1\) are drawn inside the hexagon. These six circular arcs join together to form a curve. Determine the area of the shaded region, both red and green, enclosed by this curve.
Red areas = 3 ( area of circular arc in each circle of 240° ) = 2* area of circle with radius =1 = 2pi
The white area is 1/2 of this = pi
Area of hexagon = 3sqrt (3) (side^2) / 2 = 3sqrt (3) 3sqrt (3) * 2^2 / 2 = 6sqrt (3)
Total of red, green areas =
2pi + 6sqrt (3) - pi =
pi + 6sqrt (3) ≈ 13.53