Area Of A Shaded Region

🔑 Key Concepts

  • The shaded region is found by subtracting the unshaded circles from the large circle.
  • The large circle has a radius of \(7.5\text{ cm}\), so its diameter is \(15\text{ cm}\).
  • Five small circles fit across the large diameter, so each small diameter is \(3\text{ cm}\).
  • Each small circle has a radius of \(1.5\text{ cm}\).
  • There are nine small circles because the horizontal and vertical groups share the center circle.
  • The final area must be expressed in square centimeters and rounded to the nearest tenth.

✏️ Worked Example - Area of Shaded Region

🧠 Math Vocabulary

  • Shaded region: The portion of a figure whose area is being measured.
  • Congruent circles: Circles that have equal radii and equal diameters.
  • Radius: A segment from the center of a circle to a point on the circle.
  • Diameter: A segment through the center with endpoints on the circle; it equals twice the radius.
  • Composite figure: A figure formed from two or more simpler shapes.
  • Repeated shape: A congruent shape that appears more than once in a diagram.
  • Area: The amount of two-dimensional space inside a figure.
  • Square unit: A unit used to measure area, such as a square centimeter.
  • Nearest tenth: A value rounded to one digit after the decimal point.

💡 Main Idea

To find the shaded area, calculate the area of the large circle and subtract the combined area of the nine smaller circles. The small radius is not given directly, so it must be determined by recognizing that five congruent circles span the large circle’s diameter. Careful counting, decimal operations, and correct use of \(A=\pi r^2\) are all important.

📚 What You Should Already Know

You should know how to use the circle area formula \(A=\pi r^2\), distinguish radius from diameter, multiply and subtract decimals, and recognize repeated congruent shapes in a diagram. You should also understand that subtracting unshaded regions from a larger figure leaves the shaded area.

🚀 What Comes Next

Next, students can solve shaded-region problems involving semicircles, quarter circles, overlapping circular regions, and combinations of circles with rectangles, squares, and other polygons.

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