https://file.aiquickdraw.com/mathbot/file/17303103963178dc7p5a1.jpg

include Pythagorean Theorem to solve and find the area   

To find the area of this shape, which is a combination of a rectangle and a half-circle, we'll use a few steps and employ the Pythagorean theorem to verify dimensions if necessary. Let's proceed as follows:

### Step 1: Find the Radius of the Half-Circle
The given dimensions indicate that the length from the left side to the right edge of the rectangle is \(7 \, \text{cm}\). Since the rectangle has a width of \(4 \, \text{cm}\), this leaves a radius of the half-circle of:
\[
r = 7 \, \text{cm} - 4 \, \text{cm} = 3 \, \text{cm}
\]

### Step 2: Calculate the Area of the Rectangle
The area \(A_{\text{rectangle}}\) of the rectangle is given by:
\[
A_{\text{rectangle}} = \text{width} \times \text{height} = 4 \, \text{cm} \times 6 \, \text{cm} = 24 \, \text{cm}^2
\]

### Step 3: Calculate the Area of the Half-Circle
The area of a full circle with radius \(r = 3 \, \text{cm}\) is:
\[
A_{\text{circle}} = \pi r^2 = 3.14 \times (3)^2 = 3.14 \times 9 = 28.26 \, \text{cm}^2
\]
Since we only have a half-circle, we divide this by 2:
\[
A_{\text{half-circle}} = \frac{28.26}{2} = 14.13 \, \text{cm}^2
\]

### Step 4: Add the Areas
Now, we add the area of the rectangle and the half-circle to find the total area of the figure:
\[
A_{\text{total}} = A_{\text{rectangle}} + A_{\text{half-circle}} = 24 \, \text{cm}^2 + 14.13 \, \text{cm}^2 = 38.13 \, \text{cm}^2
\]

### Answer
The total area enclosed by the figure is:
\[
\boxed{38.13 \, \text{cm}^2}
\]

Would you like further details or have any other questions?

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### Relative Questions:
1. How would the area change if the radius of the half-circle were different?
2. How is the Pythagorean theorem used in verifying the dimensions of composite shapes?
3. What if the shape was a full circle instead of a half-circle?
4. How do different values of \( \pi \) affect area calculations?
5. Can this method be generalized for other composite shapes?

### Tip:
When dealing with composite shapes, break down each part, calculate their areas individually, and then sum them to find the total area.

Find the area enclosed by the figure. Use 3.14 for π and do not round your answer. The figure is not to scale.

Learn how to calculate the area of a composite shape made up of a rectangle and a half-circle. The problem uses 3.14 for π and covers key geometric formulas like the area of a rectangle and the area of a half-circle. Suitable for Grade 6-8 students.

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