Problem Breakdown:

We are tasked with calculating the total area of the two shaded regions in terms of $\pi$.

Key Details:

1. Each circle has a radius of 4 cm.
2. The centers $A$, $B$, and $C$ are collinear, with $AB = BC = 4 \, \text{cm}$.
3. This configuration results in three identical circles that intersect as shown, where:
   • $A$ and $B$'s circles overlap, forming one shaded region.
   • $B$ and $C$'s circles overlap, forming another identical shaded region.

Step 1: Understanding the Overlap of Two Circles

The shaded region between two intersecting circles is called a lens or a lune.

For two identical circles intersecting such that the distance between their centers equals the radius $r$, the area of the overlap is given by:

$A_{\text{overlap}} = 2 \left[ r^2 \cos^{-1}\left(\frac{d}{2r}\right) - \frac{d}{2} \sqrt{4r^2 - d^2} \right]$

Where:

• $r = 4 \, \text{cm}$ (radius of the circles),
• $d = 4 \, \text{cm}$ (distance between centers $A$ and $B$, or $B$ and $C$).

Step 2: Substituting Values

1. Cosine Term: $\cos^{-1}\left(\frac{d}{2r}\right) = \cos^{-1}\left(\frac{4}{8}\right) = \cos^{-1}\left(\frac{1}{2}\right)$ Using standard trigonometric values: $\cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3}$

2. Square Root Term: $\sqrt{4r^2 - d^2} = \sqrt{4(4)^2 - (4)^2} = \sqrt{64 - 16} = \sqrt{48} = 4\sqrt{3}$

3. Substitute into the formula for the area of one shaded region: $A_{\text{overlap}} = 2 \left[ (4^2) \cdot \frac{\pi}{3} - \frac{4}{2} \cdot 4\sqrt{3} \right]$ Simplify each term:

   • $(4^2) \cdot \frac{\pi}{3} = \frac{16\pi}{3}$
   • $\frac{4}{2} \cdot 4\sqrt{3} = 8\sqrt{3}$

   So:

   = \frac{32\pi}{3} - 16\sqrt{3}$$

Step 3: Total Area of Two Shaded Regions

Since the two shaded regions are identical, multiply by 2: $A_{\text{total}} = 2 \cdot \left(\frac{32\pi}{3} - 16\sqrt{3}\right) = \frac{64\pi}{3} - 32\sqrt{3}$

Final Answer:

The total area of the two shaded regions is: $\boxed{\frac{64\pi}{3} - 32\sqrt{3} \, \text{cm}^2}$

Would you like me to expand on the derivation or clarify any part of the solution?
Here are 5 questions to help deepen your understanding:

1. How is the overlap area formula derived for intersecting circles?
2. Why does $\cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3}$?
3. What happens to the shaded region if the distance between centers changes?
4. Can this approach be extended to circles with different radii?
5. How can the result be approximated numerically without $\pi$?

Tip: Familiarize yourself with trigonometric identities like $\cos^{-1}(x)$, as they often appear in geometry problems!