To solve this problem, we'll determine the area of the shaded region between the two concentric circles, given that the circumference of the inner circle is 31.4 centimeters, and the radius of the outer circle is 2 centimeters.

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 $31.4$ centimeters, we can solve for the radius $r$:

$31.4 = 2\pi r$

$r = \frac{31.4}{2\pi} = \frac{31.4}{6.28} = 5 \text{ centimeters}$

So, the radius of the inner circle is $5$ centimeters.

Step 2: Determine the radius of the outer circle.

The problem states that the outer circle has a radius of $2$ centimeters. However, this doesn't make sense if the inner circle has a radius of 5 centimeters. The outer circle should have a larger radius than the inner circle.

If the radius of the outer circle is intended to be $7$ centimeters (which would be 2 centimeters more than the radius of the inner circle), let's proceed with that assumption.

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: $A_{\text{inner}} = \pi \times (5)^2 = \pi \times 25 = 78.5 \text{ square centimeters}$

• Outer circle area: If the outer circle's radius is 7 centimeters: $A_{\text{outer}} = \pi \times (7)^2 = \pi \times 49 = 153.86 \text{ square centimeters}$

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:

$A_{\text{shaded}} = A_{\text{outer}} - A_{\text{inner}} = 153.86 - 78.5 = 75.36 \text{ square centimeters}$

Final Answer:

The area of the shaded region is 75.36 square centimeters.

Would you like more details or have any questions?

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Here are 5 related questions for you to explore:

1. How do you calculate the circumference of a circle if you know its diameter?
2. What is the relationship between the circumference and the area of a circle?
3. How can you find the area of an annulus (the region between two concentric circles)?
4. What happens to the area of a circle if the radius is tripled?
5. How do you determine the radius of a circle given its area?

Tip: Always double-check the given values in a problem, especially if they seem inconsistent, to ensure you're interpreting the problem correctly.