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find the area of the overlap between the two circles

To find the area of overlap between the two circles, we can apply a geometric approach based on their radii and the distance between their centers.

### Given:
- Radius of the smaller circle, \( r_1 = 3.6 \, \text{cm} \)
- Radius of the larger circle, \( r_2 = 4.8 \, \text{cm} \)
- Distance between the centers of the circles, \( d = 6 \, \text{cm} \)

### Steps:
The area of overlap between two intersecting circles can be found using a formula derived from geometry and trigonometry. Specifically, the overlap area \( A \) is the sum of the areas of the two circular segments, which can be calculated as:

\[
A = r_1^2 \cos^{-1} \left( \frac{d^2 + r_1^2 - r_2^2}{2 d r_1} \right) + r_2^2 \cos^{-1} \left( \frac{d^2 + r_2^2 - r_1^2}{2 d r_2} \right) - \frac{1}{2} \sqrt{(-d + r_1 + r_2)(d + r_1 - r_2)(d - r_1 + r_2)(d + r_1 + r_2)}
\]

Where:
- \( r_1 \) and \( r_2 \) are the radii of the two circles.
- \( d \) is the distance between the centers of the circles.
- The term involving square root calculates the area of the triangle formed by the centers and the intersection points.

### Plugging in the values:
1. \( r_1 = 3.6 \, \text{cm}, r_2 = 4.8 \, \text{cm}, d = 6 \, \text{cm} \)
   
Now we will calculate this step by step:

#### 1. Calculate the angles of the circular segments using the inverse cosine function.

\[
\theta_1 = 2 \cos^{-1} \left( \frac{6^2 + 3.6^2 - 4.8^2}{2 \cdot 6 \cdot 3.6} \right)
\]
\[
\theta_2 = 2 \cos^{-1} \left( \frac{6^2 + 4.8^2 - 3.6^2}{2 \cdot 6 \cdot 4.8} \right)
\]

#### 2. Calculate the area of each circular segment.

\[
A_1 = \frac{1}{2} r_1^2 \left( \theta_1 - \sin(\theta_1) \right)
\]
\[
A_2 = \frac{1}{2} r_2^2 \left( \theta_2 - \sin(\theta_2) \right)
\]

#### 3. Finally, sum the areas of the two segments to get the overlap area.

Let me compute the exact value for you.The area of overlap between the two circles is approximately \( 9.56 \, \text{cm}^2 \).

Would you like further details on this calculation or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do the radii of the circles affect the overlap area?
2. What would happen to the overlap if the distance between the centers increased or decreased?
3. How can the formula for the area of circular segments be derived geometrically?
4. How does the overlap area change if both circles had the same radius?
5. Can this approach be extended to three or more overlapping circles?

**Tip:** The cosine inverse function helps determine the angles of intersection, which are critical in calculating the area of circular segments.

Find the area of overlap in the two circles below. The smaller circle has a radius of 3.6 cm, and the larger circle has a radius of 4.8 cm. The distance between the centers of the circles is 6 cm.

Learn how to find the area of overlap between two intersecting circles with radii 3.6 cm and 4.8 cm. This geometry problem involves circular segments, trigonometric calculations, and the use of the law of cosines. Explore a detailed step-by-step approach to solve this problem, ideal for Grades 10-12.

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