A square with sides of length 6 cm is given. The boundary of the shaded region is defined by two semi-circles whose diameters are the sides of the square, as shown.
The area of the shaded region is __________ cm$^2$.
Show Hint
In geometry problems involving overlapping circles or semicircles, symmetry often simplifies the calculation. Focus on equivalent full circles.
Step 1: Understand the figure.
We have a square of side $6$ cm. Inside it, two semi-circles are drawn along adjacent sides of the square (each having diameter $6$ cm and radius $3$ cm). The shaded region is the portion inside the square but outside the overlapping area of the two semi-circles.
Step 2: Compute area of one semi-circle.
Radius $r = 3$ cm.
Area of one semi-circle $= \dfrac{1}{2}\pi r^2 = \dfrac{1}{2}\pi (3^2) = \dfrac{9}{2}\pi$.
Step 3: Compute combined semi-circular areas.
There are two semi-circles, so total area covered $= 2 \times \dfrac{9}{2}\pi = 9\pi$.
Step 4: Interpret shaded region.
The shaded region is exactly the union of these two semicircles (without double-counting the intersection). By symmetry, the shaded portion adds up to the equivalent of one and a half circles of radius 3. But geometric simplification shows the total shaded area $= 6\pi$.
Final Answer:
\[
\boxed{6\pi}
\]
