To find the area of the shaded region around a semicircle placed inside a rectangle, follow these steps:

1. Determine the dimensions of the semicircle.
2. Calculate the area of the rectangle.
3. Calculate the area of the semicircle.
4. Subtract the area of the semicircle from the area of the rectangle to find the shaded region.

Step 1: Dimensions of the Semicircle

Given the width of the rectangle is 6 ft, this is also the diameter of the semicircle.

• Radius of the semicircle ($r$) = Diameter / 2 = $6 \, \text{ft} / 2 = 3 \, \text{ft}$

Step 2: Area of the Rectangle

To determine the height of the rectangle, consider the height as equal to the radius of the semicircle, since the semicircle fits perfectly within the rectangle.

• The width of the rectangle ($w$) = 6 ft
• The height of the rectangle ($h$) = 3 ft (radius of the semicircle)

The area of the rectangle ($A_{\text{rectangle}}$) is given by: $A_{\text{rectangle}} = w \times h$ $A_{\text{rectangle}} = 6 \, \text{ft} \times 3 \, \text{ft}$ $A_{\text{rectangle}} = 18 \, \text{ft}^2$

Step 3: Area of the Semicircle

The area of the semicircle ($A_{\text{semicircle}}$) is half the area of a full circle with radius $r$: $A_{\text{semicircle}} = \frac{1}{2} \times \pi r^2$ $A_{\text{semicircle}} = \frac{1}{2} \times 3.14 \times (3 \, \text{ft})^2$ $A_{\text{semicircle}} = \frac{1}{2} \times 3.14 \times 9 \, \text{ft}^2$ $A_{\text{semicircle}} = \frac{1}{2} \times 28.26 \, \text{ft}^2$ $A_{\text{semicircle}} = 14.13 \, \text{ft}^2$

Step 4: Area of the Shaded Region

The area of the shaded region is the area of the rectangle minus the area of the semicircle: $A_{\text{shaded}} = A_{\text{rectangle}} - A_{\text{semicircle}}$ $A_{\text{shaded}} = 18 \, \text{ft}^2 - 14.13 \, \text{ft}^2$ $A_{\text{shaded}} = 3.87 \, \text{ft}^2$

So, the area of the shaded region is $3.87 \, \text{ft}^2$.

Would you like further details or have any questions?

Questions:

1. How do you calculate the radius of a semicircle given its diameter?
2. What is the formula for the area of a rectangle?
3. How do you determine the area of a semicircle?
4. Why is it important to correctly interpret the dimensions of geometric shapes?
5. What is the relationship between the diameter and the radius of a circle?
6. How do you calculate the area of a circle?
7. Why do you subtract the area of the semicircle from the area of the rectangle to find the shaded region?
8. What unit is used to express the area of a region?

Tip:

Always double-check the dimensions and relationships of geometric shapes to ensure accurate area calculations.