The problem provided involves finding the exact perimeter of composite shapes. Let's analyze and solve the two composite shapes in the image.

(a) First Composite Shape:

This shape is a rectangle combined with a semicircle.

• Rectangle: The rectangle has a length of 6 cm and a height of 4 cm.
• Semicircle: The semicircle is attached to the top of the rectangle, with its diameter equal to the width of the rectangle, which is 6 cm.

Steps to calculate the perimeter:

1. Perimeter of the rectangle (excluding the side attached to the semicircle):

   • $P_{\text{rectangle}} = 6 \, \text{cm} + 4 \, \text{cm} + 4 \, \text{cm} = 14 \, \text{cm}$ (we exclude the top side)

2. Circumference of the semicircle:

   • The diameter is 6 cm, so the radius $r = \frac{6}{2} = 3 \, \text{cm}$.
   • Circumference of a full circle is $2\pi r$, so the circumference of the semicircle is $\pi r = \pi \times 3 \, \text{cm} = 3\pi \, \text{cm}$.

3. Total Perimeter:

   • $P_{\text{total}} = 14 \, \text{cm} + 3\pi \, \text{cm}$.
   • The exact perimeter is $14 + 3\pi$ cm.

(b) Second Composite Shape:

This shape is a rectangle with a semicircle on one side.

• Rectangle: The rectangle has a width of 5 cm and a height of 2 cm.
• Semicircle: The semicircle is attached to the top of the rectangle, with its diameter equal to the width of the rectangle, which is 5 cm.

Steps to calculate the perimeter:

1. Perimeter of the rectangle (excluding the side attached to the semicircle):

   • $P_{\text{rectangle}} = 5 \, \text{cm} + 2 \, \text{cm} + 2 \, \text{cm} = 9 \, \text{cm}$ (we exclude the top side)

2. Circumference of the semicircle:

   • The diameter is 5 cm, so the radius $r = \frac{5}{2} = 2.5 \, \text{cm}$.
   • Circumference of a full circle is $2\pi r$, so the circumference of the semicircle is $\pi r = \pi \times 2.5 \, \text{cm} = 2.5\pi \, \text{cm}$.

3. Total Perimeter:

   • $P_{\text{total}} = 9 \, \text{cm} + 2.5\pi \, \text{cm}$.
   • The exact perimeter is $9 + 2.5\pi$ cm.

Summary of Results:

• For the first shape (a), the exact perimeter is $14 + 3\pi$ cm.
• For the second shape (b), the exact perimeter is $9 + 2.5\pi$ cm.

Would you like more details on these calculations, or do you have any other questions?

Relative Questions:

1. What is the formula for the circumference of a full circle?
2. How would you find the area of these composite shapes instead of the perimeter?
3. Can you explain how the perimeter of a rectangle is generally calculated?
4. What is the significance of using exact values like $\pi$ in these perimeter calculations?
5. How would the perimeter change if the shapes had a full circle instead of a semicircle?

Tip:

When working with composite shapes, it's important to carefully consider which parts of the shape contribute to the total perimeter.