The image shows a composite shape consisting of a rectangle and a semicircle on top of it. The given dimensions are 34 cm for the height of the rectangle, which is also the diameter of the semicircle.

1. Calculating the Perimeter:

The perimeter consists of:

• The two sides of the rectangle (excluding the base that aligns with the diameter of the semicircle)
• The curved edge of the semicircle.

Let's break it down:

• Two vertical sides of the rectangle: $2 \times 34$ cm
• Circumference of the semicircle: $\frac{1}{2} \times \pi \times \text{Diameter}$ $\text{Diameter} = 34 \text{ cm}$ $\text{Circumference of semicircle} = \frac{1}{2} \times \pi \times 34 \text{ cm}$

So, the perimeter $P$ is: $P = 2 \times 34 + \frac{1}{2} \times \pi \times 34$

2. Calculating the Area:

The area consists of:

• The area of the rectangle
• The area of the semicircle.

Let's break it down:

• Area of the rectangle: $\text{Length} \times \text{Width}$ $\text{Length} = 34 \text{ cm (same as the diameter)}$ $\text{Width} = 34 \text{ cm}$ $\text{Area of rectangle} = 34 \times 34 \text{ cm}^2$
• Area of the semicircle: $\frac{1}{2} \times \pi \times \left(\frac{34}{2}\right)^2$

So, the area $A$ is: $A = 34 \times 34 + \frac{1}{2} \times \pi \times \left(\frac{34}{2}\right)^2$

Now, let's compute the exact values:

1. Perimeter: $P = 2 \times 34 + \frac{1}{2} \times \pi \times 34 \approx 68 + 53.41 \approx 121.41 \text{ cm}$

2. Area: $A = 34 \times 34 + \frac{1}{2} \times \pi \times 17^2 \approx 1156 + 453.645 \approx 1609.645 \text{ cm}^2$

So, the perimeter is approximately 121.41 cm, and the area is approximately 1609.65 cm².

Would you like more details or have any questions?

Here are some relative questions:

1. How would the perimeter change if the radius of the semicircle were doubled?
2. How is the formula for the area of a semicircle derived from the area of a full circle?
3. What is the relationship between the diameter and radius of a circle, and how does it affect the calculations?
4. Can the method used here apply to shapes other than semicircles? If so, how?
5. How would the calculations change if the shape were a quarter-circle rather than a semicircle?

Tip: When working with composite shapes, break them down into simpler geometric figures to simplify calculations.