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The problem you've provided is a geometry question involving two intersecting circles, each with a radius of 6 units, and a central angle of \(60^\circ\) between the centers of the circles, \(A\) and \(B\). The task is to find the area of the shaded region where the circles overlap.

Here’s a step-by-step breakdown of how to solve this:

### Step 1: Area of a sector
Each circle has a radius \(r = 6\), and the angle of the sector for each circle is \(60^\circ\), or \(\frac{\pi}{3}\) radians.

The formula for the area of a sector is:
\[
A_{\text{sector}} = \frac{\theta}{2\pi} \times \pi r^2 = \frac{\theta}{360^\circ} \times \pi r^2
\]
For each sector (where \(\theta = 60^\circ\)):
\[
A_{\text{sector}} = \frac{60}{360} \times \pi (6)^2 = \frac{1}{6} \times 36\pi = 6\pi
\]
Thus, the area of each sector is \(6\pi\).

### Step 2: Area of the triangle formed by the two radii and the chord
The two radii of the circles and the chord form an equilateral triangle with side length \(r = 6\). The area of an equilateral triangle is given by:
\[
A_{\text{triangle}} = \frac{\sqrt{3}}{4} s^2
\]
where \(s\) is the side length of the triangle, in this case 6:
\[
A_{\text{triangle}} = \frac{\sqrt{3}}{4} \times 6^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3}
\]
So, the area of the equilateral triangle is \(9\sqrt{3}\).

### Step 3: Area of the segment
The area of each segment (shaded region inside the sector but outside the triangle) is:
\[
A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} = 6\pi - 9\sqrt{3}
\]

### Step 4: Total shaded area
Since there are two such segments (one for each circle), the total shaded area is:
\[
A_{\text{total}} = 2 \times (6\pi - 9\sqrt{3}) = 12\pi - 18\sqrt{3}
\]

Thus, the area of the shaded region is:

\[
\boxed{12\pi - 18\sqrt{3}}
\]

### Answer: \(12\pi - 18\sqrt{3}\).

Let me know if you'd like more details or explanations!

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Here are 5 related questions for further exploration:
1. How do you calculate the area of a sector with different angles?
2. What is the significance of the central angle in circle geometry problems?
3. How do you find the area of the shaded region when the angle between two radii is not \(60^\circ\)?
4. What would the area be if the radius of the circles were doubled but the angle stayed the same?
5. How would this problem change if the circles had different radii?

**Tip:** For circle problems involving intersections, always separate the calculation into sectors and triangles to simplify the area analysis.

In the diagram above, A & B are the centers of the two circles, each with radius r = 6, and ∠A = ∠B = 60°. What is the area of the shaded region?

Solve a geometry problem involving two intersecting circles, each with a radius of 6 and a 60° central angle. Learn how to calculate the area of the shaded region using sector and triangle area formulas. Suitable for high school students in Grade 10-12.

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