🔑 Key Concepts
- Find the area of the entire \(24\)-inch square.
- Two congruent circles fit across each side of the square.
- Each circle has a diameter of \(12\) inches and a radius of \(6\) inches.
- Use \(A=\pi r^2\) to find the area of one circle.
- Multiply one circle’s area by \(4\).
- Subtract the combined circle area from the square’s area.
💡 Main Idea
The shaded region is the part of the \(24\)-inch square that is not covered by the four congruent circles. Since two circles fit across each side of the square, each circle has a diameter of \(12\) inches and a radius of \(6\) inches. Find the square’s area, calculate the combined area of the four circles, and subtract.
✏️ Worked Example - Calculating Area of a Shaded Region
🧠 Math Vocabulary
- Shaded region: The portion of a figure whose area is being measured.
- Square: A quadrilateral with four congruent sides and four right angles.
- Congruent circles: Circles with equal radii and equal diameters.
- Radius: A segment from the center of a circle to its edge.
- Diameter: A segment through the center of a circle with endpoints on the circle.
- Composite figure: A figure formed by combining or removing simpler shapes.
- Area: The amount of two-dimensional space inside a figure.
- Square inch: A unit used to measure area.
📚 What You Should Already Know
You should know how to find the area of a square using \(A=s^2\), find the area of a circle using \(A=\pi r^2\), and distinguish between a circle’s radius and diameter. You should also be comfortable multiplying and subtracting decimals.
🚀 What Comes Next
Next, students can solve more advanced shaded-region problems involving semicircles, quarter circles, repeated circular cutouts, and composite figures made from several different polygons and circular regions.
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