Finding the Area of a Circle with Inscribed Squares

Key Concepts

• Area of Squares and Circle: Understanding the relationship between squares inscribed in a circle.
• Equations of a Circle: Utilizing standard circle equations to solve for unknowns.
• Coordinate System: Applying a rectangular coordinate system to identify points on a circle.

Circle and Squares Overview

1. Identify Square Properties:

   • Each square's area = 16.
   • Side length of each square = 4.

2. Circle Equations:

   • Area of a circle: A = πr²
   • Equation of a circle: (x - h)² + (y - k)² = r²
     • where (h, k) is the center and r is the radius.

Steps to Determine the Circle's Area

1. Identify Key Points on Circle: Four Inscribed Squares Touching Points

   • Coordinate system used to identify points.
   • Points identified: (0, 0), (0, 4), (-16, -4)

2. Solve for Variables in Circle Equation:

   • Equations derived from points:
     1. (0 - h)² + (0 - k)² = r²
     2. (0 - h)² + (4 - k)² = r²
     3. (-16 - h)² + (-4 - k)² = r²
   • Simplify and solve for h, k, and r systematically.

3. Elimination and Simplification: Solving for k

   • Starting with two top equations:
     • h² + k² = h² + (4 - k)²
     • Simplify: k² = 16 - 8k + k²
     • Result: k = 2.

4. Solve for Remaining Variables: Solving for h

   • Substitute k = 2 back into equations.
     • (4 - k)² simplifies to 2² = 4
     • -4 - k simplifies to -6² = 36
   • Solve the simplified equations:
     • -16 - h simplifies to (-16 + 9)² = 81
     • Result: h = -9.

Final Calculation of Radius and Area

1. Substitute h and k into Equations

   • Validate by confirming consistency in equations.
   • Resultant r² = 85

2. Calculate Circle's Area:

   • A = π × 85
     • Result: Circle's area is 85π square units.

Conclusion

• The area of the circle with inscribed squares is 85π square units.