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To find the perimeter and area of the given figure, we will treat it as a combination of a rectangle and a half-circle.

### **Step-by-Step Solution:**

#### **1. Dimensions from the figure:**
- The length of the rectangle is **12 units**.
- The width of the rectangle is **10 units**.
- The diameter of the half-circle is the same as the width of the rectangle, which is **10 units**.
- Therefore, the radius of the half-circle is **5 units** (since the radius is half of the diameter).

#### **2. Calculating the Area:**

- **Area of the Rectangle:**
  \[
  \text{Area}_{\text{rectangle}} = \text{Length} \times \text{Width} = 12 \times 10 = 120 \, \text{square units}
  \]

- **Area of the Half-Circle:**
  \[
  \text{Area}_{\text{half-circle}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (5)^2 = \frac{1}{2} \pi \times 25 = 12.5 \pi \, \text{square units}
  \]

- **Total Area:**
  \[
  \text{Total Area} = \text{Area}_{\text{rectangle}} + \text{Area}_{\text{half-circle}} = 120 + 12.5 \pi
  \]
  \[
  \text{Total Area} \approx 120 + 39.27 = 159.27 \, \text{square units}
  \]
  (Using \(\pi \approx 3.14\))

#### **3. Calculating the Perimeter:**

- **Perimeter of the Rectangle (excluding the straight side along the diameter of the half-circle):**
  The perimeter will consist of three sides of the rectangle and the curved edge of the half-circle.
  - The two vertical sides (both of **10 units**) and the bottom side (**12 units**) give:
  \[
  10 + 10 + 12 = 32 \, \text{units}
  \]

- **Curved Edge of the Half-Circle:**
  \[
  \text{Circumference of the full circle} = 2 \pi r = 2 \pi \times 5 = 10 \pi
  \]
  \[
  \text{Curved part of the half-circle} = \frac{1}{2} \times 10 \pi = 5 \pi \, \text{units}
  \]

- **Total Perimeter:**
  \[
  \text{Total Perimeter} = 32 + 5 \pi
  \]
  \[
  \text{Total Perimeter} \approx 32 + 15.7 = 47.7 \, \text{units}
  \]
  (Using \(\pi \approx 3.14\))

### **Final Answers:**
- **Area of the figure:** \( \approx 159.27 \, \text{square units} \)
- **Perimeter of the figure:** \( \approx 47.7 \, \text{units} \)

Would you like a detailed explanation of any specific step, or do you have any questions?

### **Related Questions:**

1. How do you calculate the perimeter of a figure that is part polygon and part curve?
2. What are the different formulas for calculating the area of a circle and parts of a circle?
3. How does changing the radius of the half-circle affect the area and perimeter?
4. Can you explain how to find the area of composite shapes involving triangles and circles?
5. What is the relationship between the diameter, radius, and circumference of a circle?

**Tip:** When dealing with composite figures, break them down into simpler shapes, calculate each area or perimeter separately, and then combine the results.

Find the perimeter and area of the figure consisting of a rectangle and a half-circle, where the rectangle has dimensions 12 units by 10 units, and the half-circle's diameter is equal to the width of the rectangle.

Learn how to calculate the area and perimeter of a composite figure made of a rectangle and a half-circle. This problem involves geometry concepts, including the formulas for the area and circumference of circles. Suitable for students in Grades 7-9.

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