Area of the Shaded Region Of A Trapezoid

🔑 Key Concepts

  • Find the area of the entire trapezoid before subtracting the semicircle.
  • The parallel bases of the trapezoid measure \(6\) units and \(14\) units.
  • The labeled \(4\)-unit segment is the semicircle’s radius, so the trapezoid’s full height is \(8\) units.
  • The trapezoid area formula is \(\displaystyle A=\frac{1}{2}(b_1+b_2)h\).
  • The semicircle area formula is \(\displaystyle A=\frac{1}{2}\pi r^2\).
  • Subtract the semicircle’s area from the trapezoid’s area to find the shaded region.

✏️ Worked Example - Area of a Shaded Region

🧠 Math Vocabulary

  • Composite figure: A figure made by combining or removing simpler geometric shapes.
  • Shaded region: The portion of a figure whose area is being measured.
  • Trapezoid: A quadrilateral with at least one pair of parallel sides.
  • Bases: The parallel sides of a trapezoid.
  • Height: The perpendicular distance between the parallel bases.
  • Semicircle: Half of a circle.
  • Radius: A segment from the center of a circle to a point on the circle.
  • Diameter: A segment through the center of a circle with endpoints on the circle; it equals twice the radius.
  • Subtract: To remove one area from another area.
  • Square unit: A unit used to measure area.

💡 Main Idea

This shaded region can be found by treating the figure as a complete trapezoid with a semicircle removed. First, use the two parallel bases and the full perpendicular height to find the trapezoid’s area. Next, use the \(4\)-unit radius to find half the area of a circle. Subtracting the semicircle’s area from the trapezoid’s area gives the area of the remaining yellow region.

📚 What You Should Already Know

You should know how to identify the parallel bases and perpendicular height of a trapezoid, use the formula \(\displaystyle A=\frac{1}{2}(b_1+b_2)h\), and find circle area using \(A=\pi r^2\). You should also understand that a semicircle has half the area of a full circle and that the diameter is twice the radius.

🚀 What Comes Next

Next, students can solve shaded-region problems involving quarter circles, multiple circular cutouts, irregular polygons, and composite figures made from several different shapes. These same decomposition and subtraction strategies also support later work with surface area and geometric modeling.

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