Area of a Shaded Region - Subtracting The Area Of Semicircles

🔑 Key Concepts

  • Begin by finding the area of the entire rectangle.
  • The two congruent semicircles combine to form one full circle.
  • The rectangle’s height is also the diameter of the circle.
  • Divide the diameter by \(2\) to find the radius.
  • Subtract the circle’s area from the rectangle’s area.
  • Express the result in square units.

💡 Main Idea

The shaded region is the part of the rectangle that remains after two semicircles are removed. Because the semicircles are congruent, they combine to form one full circle. First find the rectangle’s area. Next, use the rectangle’s height as the circle’s diameter and divide by \(2\) to find the radius. Find the circle’s area and subtract it from the rectangle. For this figure, the rectangle measures \(8\) units by \(4\) units, and the equivalent circle has a radius of \(2\) units.

✏️ Worked Example - Calculating Area of a Shaded Region

🧠 Math Vocabulary

  • Shaded region: The portion of a figure whose area is being measured.
  • Semicircle: Half of a circle.
  • Congruent: Having the same size and shape.
  • 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 by combining or removing simpler geometric shapes.
  • Area: The amount of two-dimensional space inside a figure.
  • Square unit: A unit used to measure area.

📚 What You Should Already Know

You should know how to find the area of a rectangle using \(A=lw\) and the area of a circle using \(A=\pi r^2\). You should also understand the relationship between radius and diameter and recognize that two congruent semicircles combine to make one complete circle.

🚀 What Comes Next

Next, students can solve shaded-region problems involving quarter circles, multiple circular cutouts, overlapping figures, and composite regions that combine circles with triangles, trapezoids, rectangles, and other polygons.

🧩 Embed This Video in Your LMS

Teachers can embed this YouTube Short directly into Canvas, Google Classroom, Schoology, or another learning platform. Click below to show the embed code.