To solve this problem, we need to calculate the area of the entire shape formed by the square and the four partial circles.

Step-by-Step Solution

1. Understanding the Figure:

   • The square has vertices at the centers of the four circles.
   • Each side of the square is $50$ units.

2. Calculate the Area of the Square:

   • The side length of the square is $50$ units, so the area of the square is: $\text{Area of square} = 50 \times 50 = 2500 \text{ square units}$

3. Calculate the Radius of Each Circle:

   • Since the vertices of the square are at the centers of the circles and the side length is $50$, the radius $r$ of each circle is also $50$ units.

4. Calculate the Area of One Circle:

   • Using $\pi = 3.1416$, the area of one circle is: $\text{Area of one circle} = \pi r^2 = 3.1416 \times 50^2 = 3.1416 \times 2500 = 7854 \text{ square units}$

5. Calculate the Area of the Four Circles Combined:

   • Since there are four circles, the total area of the circles is: $\text{Total area of circles} = 4 \times 7854 = 31416 \text{ square units}$

6. Combine Areas to Find the Total Area of the Shape:

   • Since the figure includes the square and the four partial circles (without overlapping), we add the area of the square and the total area of the circles: $\text{Total area of the shape} = 2500 + 31416 = 33916 \text{ square units}$

Answer

The closest answer to our calculation is (D) 33,562 square units.

Would you like more detailed steps on any part of the solution or further explanation?

Related Questions

1. How would the answer change if the side length of the square was different?
2. What if the square was replaced with a rectangle of different dimensions?
3. How would this calculation differ if the circles were only quarter-circles instead of full circles?
4. Can we generalize this approach for any number of circles around a central polygon?
5. What happens if the circles overlap?

Tip

When solving geometry problems, breaking the figure into familiar shapes (like circles and squares) often simplifies area calculations.