Let's solve this problem by finding the perimeter and area of the composite shape.

1. Perimeter Calculation

The perimeter is the sum of the lengths of all outer edges. This shape consists of:

• Two vertical sides of $6 \, \text{cm}$ each.
• Two horizontal sides of $9 \, \text{cm}$ each.
• A semicircular arc with radius $3 \, \text{cm}$ (since the width of the rectangle is 6 cm, the diameter of the semicircle is also 6 cm, giving a radius of $3 \, \text{cm}$).

Semicircle Arc Length

The formula for the circumference of a circle is: $C = 2\pi r$ The arc length of the semicircle is half the circumference: $\text{Arc Length} = \pi r = \pi \times 3 = 3\pi \, \text{cm}.$

Total Perimeter

Adding all edges: $\text{Perimeter} = 6 + 6 + 9 + 9 + 3\pi.$ Substitute $\pi \approx 3.14$: $\text{Perimeter} \approx 6 + 6 + 9 + 9 + 3(3.14) = 30 + 9.42 = 39.42 \, \text{cm}.$

2. Area Calculation

The area of the shape can be computed as:

• Area of the rectangle ($9 \, \text{cm} \times 6 \, \text{cm}$).
• Subtract the area of the semicircle ($\frac{1}{2} \pi r^2$).

Rectangle Area

$\text{Area of Rectangle} = 9 \times 6 = 54 \, \text{cm}^2.$

Semicircle Area

$\text{Area of Semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (3)^2 = \frac{1}{2} \pi \times 9 = 4.5\pi.$ Substitute $\pi \approx 3.14$: $\text{Area of Semicircle} \approx 4.5 \times 3.14 = 14.13 \, \text{cm}^2.$

Total Area

$\text{Area} = \text{Area of Rectangle} - \text{Area of Semicircle}.$ $\text{Area} \approx 54 - 14.13 = 39.87 \, \text{cm}^2.$

Final Answer:

• Perimeter: $\approx 39.42 \, \text{cm}$
• Area: $\approx 39.87 \, \text{cm}^2$

Would you like further clarifications or more details?

Related Questions:

1. How is the arc length of a circle calculated?
2. How would the answer change if the semicircle were replaced with a full circle?
3. What is the importance of $\pi$ in circle-related calculations?
4. Can you explain how the formula for the circumference of a circle is derived?
5. What happens to the area calculation if the semicircle is shifted to another side of the rectangle?

Tip:

When solving geometry problems with composite shapes, break the problem into smaller parts and handle one component at a time.