1. **Stating the problem:** We have a square with side length 8 units and four quarter circles of radius 8 units centered at each corner of the square. We want to understand the shape formed by the intersection of these quarter circles inside the square. 2. **Understanding the geometry:** Each quarter circle is part of a circle with radius 8 units. Since the square's side length is also 8 units, each quarter circle exactly fits into a corner of the square. 3. **Equation of the circles:** The equation given is $$x^2 + y^2 = 4^2$$ which means a circle of radius 4 units centered at the origin. However, the problem states the radius is 8 units, so the correct circle equation for each quarter circle is $$x^2 + y^2 = 8^2 = 64$$. 4. **Positioning the quarter circles:** The four quarter circles are centered at the four corners of the square: - Bottom-left corner at (0,0) - Bottom-right corner at (8,0) - Top-left corner at (0,8) - Top-right corner at (8,8) Each quarter circle covers the inside of the square near its corner. 5. **Finding the red region:** The red region is the intersection of all four quarter circles inside the square. This region is symmetrical and bounded by the arcs of the quarter circles. 6. **Summary:** The problem involves understanding the intersection of four quarter circles of radius 8 units inside an 8x8 square, forming a symmetrical shape in the center. Final note: The original equation $$x^2 + y^2 = 4^2$$ corresponds to a circle of radius 4, but the problem's geometry uses radius 8, so the correct circle equation is $$x^2 + y^2 = 64$$.