B3 In the figure, two circles with radius 6 are drawn. The centers of the circles are also 6 units apart.

Determine the area of the shaded region, which is bounded by two circular arcs and a common tangent.

![](https://cdn.mathpix.com/cropped/2024_04_17_2582a2b087f6953c1edag-2.jpg?height=363&width=537&top_left_y=721&top_left_x=1316)
B3. To find the area of the shaded region $B C S$, we subtract the area of triangle $A M S$ and the area of the circular sector $M C S$ from the rectangle $A M C B$ with area $3 \times 6$. $S A=\sqrt{6^{2}-3^{2}}=3 \sqrt{3}$. Furthermore, $\angle C M S=30^{\circ}$ because $\angle S M A=60^{\circ}$. For the shaded area, we find: $18-4 \frac{1}{2} \sqrt{3}-\frac{1}{12} \pi 6^{2}$. The required area is twice as large, so $36-9 \sqrt{3}-6 \pi$.

![](https://cdn.mathpix.com/cropped/2024_04_17_7fa3f9281a1c56596c60g-2.jpg?height=620&width=568&top_left_y=1369&top_left_x=1121)