To find the area of the shaded region in an annular shape (a ring), we use the given formula:

$A = \frac{1}{4} \pi (D^2 - d^2)$

where:

• $D$ is the diameter of the outer circle
• $d$ is the diameter of the inner circle

Part 1: Determine $D$, the diameter of the outer circle.

From the question, the information provided includes:

• Outer diameter $D = 7$ inches
• Inner diameter $d = 3.5$ inches (or $3 \, \text{and} \, \frac{1}{2}$ inches)

So, for this problem: $D = 7 \text{ inches}$ $d = 3.5 \text{ inches}$

Part 2: Substitute these values into the area formula.

1. Square the diameters: $D^2 = 7^2 = 49$ $d^2 = (3.5)^2 = 12.25$

2. Calculate $D^2 - d^2$: $D^2 - d^2 = 49 - 12.25 = 36.75$

3. Apply the formula: $A = \frac{1}{4} \pi (36.75)$ $A = 9.1875 \pi \, \text{square inches}$

Therefore, the area of the shaded region is: $A \approx 9.1875 \pi \text{ square inches}$

Would you like more detailed steps or have any questions?

Related Questions

1. How would you calculate the area if only the radius was given instead of the diameter?
2. How would the area change if the outer diameter increased by 1 inch?
3. What would the area of the shaded region be if the inner circle had a diameter of 4 inches?
4. How can you convert this area to approximate decimal form by using $\pi \approx 3.1416$?
5. How would you find the circumference of the outer and inner circles?

Tip

Remember, for annular regions, the shaded area is just the area difference between the two circles.