Constructing A Crescent Moon

Relevant wiki: Length and Area - Composite Figures - Intermediate

There is no need to use trigonometry. Instead, we make the following observation:

$\text{Shaded Area} = \text{Sector}\space ACB + \text{Semicircle} \space AEC - \text{Semicircle} \space ADB$

$\text{Since Semicircle}\space ADB = \text{Semicircle} \space AEC, \text{we can simplify the equation above to,}$

$\begin{aligned} \text{Shaded Area} & = \text{Sector} \space ACB + \text{Semicircle} \space AEC - \text{Semicircle} \space AEC \\ & =\text{Sector} \space ACB \\ & =\frac{1}{6} \pi \space 6^2 \\ & = 6 \pi \end{aligned}$

FINAL STEP:

$\large \frac{\text{Shaded Area}}{\pi} = \frac{6\pi}{\pi} = \color{#D61F06}{\boxed{6}}$

AC if it rotates a full circle (360 degrees) will create a circle radius 6. If we sweep only 60 degrees the area must be one sixth of the circle so one sixth of pi x 6 squared.

If the orange and white areas are viewed as layers, semicircles ADB and AEC cancel.

This leaves 1/6 of a circle (360/60 = 1/6) for area remaining.

Te area of the two semicircles (ADB and AEC) must be equal, because they have the same diameter. The total area is the sum of the 1/6 circle ABC and the semicircle AEC subtracted by the semicircle ADB (ABC+AEC-ADB) Since the two semicircles have the same area, their sum makes zero. The total area you must find is just the area of ABC. It is 1/6th of a circle with a diameer of 6 cm. It makes 36 pi/6 = 6 pi square cm :)