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.