https://file.aiquickdraw.com/mathbot/file/1730309396571hrlp200e.jpg

Include Pythagorean Theorem formula and find the perimeter using pie formula   

Let's start by calculating the area enclosed by the shape, which consists of a rectangle and a half-circle on the right side. I'll then outline how to find the perimeter as requested.

### 1. Area Calculation

#### Step 1: Area of the Rectangle
The rectangle has a length of 9 cm and a height of 6 cm.

\[
\text{Area of Rectangle} = \text{length} \times \text{height} = 9 \, \text{cm} \times 6 \, \text{cm} = 54 \, \text{cm}^2
\]

#### Step 2: Area of the Half-Circle
The diameter of the half-circle is the same as the height of the rectangle, which is 6 cm. This makes the radius \( r \) of the half-circle 3 cm (since radius = diameter / 2).

Using \( \pi = 3.14 \):
\[
\text{Area of Half-Circle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \times 3.14 \times (3)^2 = \frac{1}{2} \times 3.14 \times 9 = 14.13 \, \text{cm}^2
\]

#### Step 3: Total Area
Now, we add the area of the rectangle and the half-circle.

\[
\text{Total Area} = 54 \, \text{cm}^2 + 14.13 \, \text{cm}^2 = 68.13 \, \text{cm}^2
\]

### 2. Perimeter Calculation

To find the perimeter, we need to calculate the lengths around the shape:
- The two vertical sides of the rectangle (6 cm each),
- The bottom side of the rectangle (9 cm),
- The semicircular arc (half the circumference of a circle with a 6 cm diameter).

#### Step 1: Perimeter of the Half-Circle
The circumference of a full circle with radius \( r = 3 \) cm is:
\[
\text{Circumference} = 2 \pi r = 2 \times 3.14 \times 3 = 18.84 \, \text{cm}
\]
Since we only have half of the circle, the arc length of the half-circle is:
\[
\text{Arc Length} = \frac{1}{2} \times 18.84 = 9.42 \, \text{cm}
\]

#### Step 2: Total Perimeter
Now, add up all the sides:
\[
\text{Perimeter} = 6 + 6 + 9 + 9.42 = 30.42 \, \text{cm}
\]

### Summary of Answers
- **Total Area**: \( 68.13 \, \text{cm}^2 \)
- **Total Perimeter**: \( 30.42 \, \text{cm} \)

Would you like further details or have any questions?

### Related Questions
1. How would the area change if the diameter of the half-circle was 8 cm instead?
2. What is the effect on the perimeter if the rectangle’s height is increased to 8 cm?
3. How would using a different value of \( \pi \) (like 3.14159) affect the area?
4. If the rectangle were rotated 90 degrees, how would that impact the shape’s area or perimeter?
5. Can you find the perimeter of a full circle with a diameter of 6 cm?

### Math Tip
When calculating areas of composite shapes, break down the shape into simpler parts, find each area separately, and then combine the results.

Find the area enclosed by the figure, which consists of a rectangle and a half-circle. Use 3.14 for π and do not round your answer.

Learn how to find the area and perimeter of a composite shape combining a rectangle and a half-circle. This problem involves geometry concepts such as area calculation for rectangles and circles, and perimeter determination. Suitable for middle school students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation