https://file.aiquickdraw.com/mathbot/file/1727452531861cm4udufh.jpg

The composite shape in the image consists of a central square and four quarter-circles at the corners. Let’s break the shape down step by step to calculate both the perimeter and the total area.

### Step 1: Central Square
- Side of the square = 8 inches
- Area of the square = \( \text{side}^2 = 8^2 = 64 \, \text{square inches} \).

### Step 2: Quarter-Circles
- Each quarter-circle has a radius of 4 inches (as indicated on the diagram).
- Area of one full circle = \( \pi \times r^2 = 3.14 \times 4^2 = 50.24 \, \text{square inches} \).
- Since each quarter-circle is \( \frac{1}{4} \) of a full circle, the area of one quarter-circle is:
  \[
  \frac{1}{4} \times 50.24 = 12.56 \, \text{square inches}
  \]
- There are 4 quarter-circles, so the total area of the circular parts is:
  \[
  4 \times 12.56 = 50.24 \, \text{square inches}
  \]

### Step 3: Total Area
- Total area of the composite shape = Area of the square + Area of the quarter-circles
  \[
  64 + 50.24 = 114.24 \, \text{square inches}
  \]

### Step 4: Perimeter
- The perimeter consists of the curved parts of the four quarter-circles. Together, these four quarter-circles form a complete circle.
- The circumference of the full circle formed by the quarter-circles is:
  \[
  \text{Circumference} = 2 \pi r = 2 \times 3.14 \times 4 = 25.12 \, \text{inches}
  \]

Thus, the perimeter of the shape is 25.12 inches.

### Conclusion:
- The correct answer is **Perimeter = 25.12 inches, Area = 114.24 square inches**. None of the provided options seem to exactly match this result, but based on the problem-solving process, this would be the accurate calculation.

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Would you like any details or further clarifications? Here are five related questions:

1. How do we handle composite shapes with more complex geometries?
2. Why does the perimeter calculation only consider the curved parts and not the straight edges?
3. How can we verify the results using an approximation of \( \pi \)?
4. How would the solution change if the central square was larger or smaller?
5. What happens to the area if the radius of the quarter-circles is doubled?

**Tip:** When dealing with composite shapes, always split them into familiar basic shapes (like circles, squares, or triangles) to simplify the calculations.

Find the perimeter and total area of the composite shape shown below. All measurements are given in inches. Use π = 3.14 in any formulas used.

Learn how to find the perimeter and area of a composite shape made up of a square and four quarter-circles. This geometry problem involves calculating both area and perimeter using basic formulas for squares and circles, suitable for Grades 7-9.

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