Math Problem Statement
Solution
We are given a diagram with four circles, each of radius , centered at points , , , and , forming a square. The task is to find the area of the shaded region, which consists of the overlapping portions of the circles inside the square.
Key Observations:
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Square and Circles Configuration:
- The square has its vertices at the centers of the circles.
- The side length of the square is equal to the diameter of one circle. Since the radius is , the diameter is . Therefore, the side of the square is .
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Shaded Region:
- The shaded region consists of the area of the square minus the areas of the segments of the circles that lie outside the square.
- Each corner of the square cuts a quarter of each circle.
Steps to Calculate the Area of the Shaded Region:
1. Area of the Square:
The area of square is:
2. Area of One Circle:
The area of one circle is:
3. Area of the Four Circular Segments Inside the Square:
Each quarter of a circle is one-fourth the area of the circle, and there are four such quarters (one from each circle), which together form a full circle. Thus, the total area of the four quarter-circles is equivalent to the area of one full circle:
4. Shaded Area:
The shaded area is the area of the square minus the area of the four quarter-circles:
Using :
Thus, the area of the shaded region is approximately 42.06 cm².
Would you like more details or have any questions?
Here are 5 related questions:
- What is the formula for the area of a circle segment?
- How would the shaded area change if the radius of the circles were doubled?
- What if the quadrilateral was not a square, but a rectangle—how would that affect the calculation?
- Could this problem be solved using polar coordinates? If so, how?
- How do we calculate the perimeter of the square formed by the centers of the circles?
Tip: When solving geometric problems involving circles and squares, always look for symmetry to simplify the area calculations.
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Math Problem Analysis
Mathematical Concepts
Geometry
Circles
Area of Square
Area of Circular Segments
Formulas
Area of square = side^2
Area of a circle = πr^2
Area of circular segment (quarter circle) = (1/4)πr^2
Theorems
Basic Geometry Theorems related to areas of squares and circles
Suitable Grade Level
Grades 9-10
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