Calculate the area of ​​the shaded region if ABCD is a square whose measure is 4m.

To calculate the area of the shaded region in the square \( ABCD \), where the side length of the square is given as 4 meters, we can proceed with the following steps:

### 1. **Area of the Square:**
   \[
   \text{Area of square} = s^2 = 4^2 = 16 \text{ m}^2
   \]

### 2. **Area of Each Quarter Circle:**
   Each quarter circle has a radius equal to the side length of the square, which is 4 meters.

   \[
   \text{Area of one quarter circle} = \frac{1}{4} \times \pi \times r^2 = \frac{1}{4} \times \pi \times 4^2 = \frac{\pi \times 16}{4} = 4\pi \text{ m}^2
   \]

### 3. **Area of the Intersection (Shaded Region):**
   The shaded region is the area where the two quarter circles intersect.

   **Step 1**: Calculate the area of the sector:
   - The area of one quarter circle is \( 4\pi \) square meters.

   **Step 2**: Calculate the area of the triangular part under the sector:
   - The area of the triangle formed by the radii of the quarter circle is:
     \[
     \text{Area of Triangle} = \frac{1}{2} \times 4 \times 4 = 8 \text{ m}^2
     \]

   **Step 3**: Calculate the area of one segment (sector minus triangle):
   - The area of one segment is:
     \[
     \text{Area of Segment} = 4\pi \text{ m}^2 - 8 \text{ m}^2
     \]

   **Step 4**: The total shaded area:
   - Since the shaded region consists of two such segments:
     \[
     \text{Shaded Area} = 16 \text{ m}^2 - 2 \times \left(4\pi \text{ m}^2 - 8 \text{ m}^2\right)
     \]
   - Simplifying:
     \[
     \text{Shaded Area} = 16 \text{ m}^2 - (8\pi \text{ m}^2 - 16 \text{ m}^2) = 32 - 8\pi \text{ m}^2
     \]

### Final Answer:
The area of the shaded region is:
\[
\boxed{(32 - 8\pi) \text{ m}^2}
\]
This evaluates to approximately:
\[
\boxed{6.87 \text{ m}^2}
\]

If you have any questions or want a further breakdown of this solution, please let me know!

**Related Questions:**
1. How do you calculate the area of a full circle?
2. What changes if we double the side length of the square?
3. What is the difference in area when the quarter circles are drawn from different corners of the square?
4. Can we solve this problem using calculus? If so, how?
5. How can we generalize this method for other polygonal shapes?

**Tip:** Remember that the constant \( \pi \) is approximately 3.14159, which is useful for estimating areas involving circles.

Learn how to calculate the area of the shaded region in a square with quarter circles. Detailed step-by-step solution provided, covering the properties of squares and circle geometry. Suitable for high school students.

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