Both circles have the same center. The circumference of the inner circle is 12.56 inches. What is the area of the shaded region?

C

=

12.56 in

3 in

To solve this problem, we need to determine the area of the shaded region between the two concentric circles.

### Step 1: Find the radius of the inner circle.
The circumference \( C \) of a circle is given by the formula:

\[
C = 2\pi r
\]

Given that the circumference of the inner circle is \( 12.56 \) inches, we can solve for the radius \( r \):

\[
12.56 = 2\pi r
\]

\[
r = \frac{12.56}{2\pi} = \frac{12.56}{6.28} = 2 \text{ inches}
\]

So, the radius of the inner circle is \( 2 \) inches.

### Step 2: Determine the radius of the outer circle.
The problem states that the outer circle has a radius of \( 3 \) inches.

### Step 3: Calculate the area of both circles.
The area \( A \) of a circle is given by the formula:

\[
A = \pi r^2
\]

- **Inner circle area**:
\[
A_{\text{inner}} = \pi \times (2)^2 = \pi \times 4 = 12.56 \text{ square inches}
\]

- **Outer circle area**:
\[
A_{\text{outer}} = \pi \times (3)^2 = \pi \times 9 = 28.26 \text{ square inches}
\]

### Step 4: Calculate the area of the shaded region.
The shaded region is the difference between the area of the outer circle and the inner circle:

\[
A_{\text{shaded}} = A_{\text{outer}} - A_{\text{inner}} = 28.26 - 12.56 = 15.7 \text{ square inches}
\]

### Final Answer:
The area of the shaded region is **15.7 square inches**.

Would you like further details or have any questions?

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Here are 5 related questions for you to explore:

1. What is the formula for finding the area of a sector in a circle?
2. How do you find the circumference of a circle given its radius?
3. What is the relationship between the diameter and the radius of a circle?
4. How do you calculate the area of a circle if only the circumference is known?
5. How does the area of a circle change if the radius is doubled?

**Tip:** When working with concentric circles, always subtract the area of the smaller circle from the larger to find the area of the ring-shaped region between them.

Learn how to calculate the area of the shaded region between two concentric circles. Understand the formulas for circumference and area of circles, and apply them to find the area of the ring-shaped region. Suitable for high school students studying geometry and circle properties.

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