Finding the Area of Composite Figures with Semicircles

Key Concepts

Rectangle Area: ( ext{Length} \times ext{Width} )
- For example, a rectangle with length 6 units and width 10 units has an area of 60 square units
Circle Area: ( \pi r^2 )
- For a semicircle, the area is half of a full circle’s area: ( \frac{1}{2} \pi r^2 )
- For a quarter circle, the area is one-quarter of a full circle’s area: ( \frac{1}{4} \pi r^2 )

Rectangle: Length = 10 units, Width = 6 units
- Area: ( 10 \times 6 = 60 ) square units
Semicircle:
- Diameter = 6 units, Radius = 3 units
- Area: ( \frac{1}{2} \pi (3)^2 = \frac{9}{2} \pi = 4.5 \pi )
Total Area:
- Exact: ( 60 + 4.5 \pi ) square units
- Approximate: Using ( \pi \approx 3.14159 ), Total Area ( \approx 74.137 ) square units

Rectangle: Length = 16 units, Width = 5 units
- Area: ( 16 \times 5 = 80 ) square units
Semicircle:
- Diameter = 10 units, Radius = 5 units
- Area: ( \frac{1}{2} \pi (5)^2 = \frac{25}{2} \pi = 12.5 \pi )
Total Area:
- Exact: ( 80 + 12.5 \pi ) square units
- Approximate: Total Area ( \approx 119.3 ) square units

Square Area: Side Length = 18 units
- Area: ( 18^2 = 324 ) square units
Quarter Circle:
- Radius = 9 units
- Area: ( \frac{1}{4} \pi (9)^2 = \frac{81}{4} \pi )
Total Area:
- Exact: ( 324 - \frac{81}{4} \pi ) square units
- Approximate: Total Area ( \approx 260.4 ) square units

Trapezoid: Split into two triangles and a rectangle
- Rectangle:
  - Length = 10 units, Width = 4 units
  - Area: ( 10 \times 4 = 40 ) square units
- Triangles:
  - Each triangle: Base = 3 units, Height = 4 units
  - Area per triangle: ( \frac{1}{2} \times 3 \times 4 = 6 ) square units
Semicircle:
- Diameter = 16 units, Radius = 8 units
- Area: ( \frac{1}{2} \pi (8)^2 = \frac{64}{2} \pi = 32 \pi )
Total Area:
- Exact: ( 40 + 6 + 6 + 32 \pi = 52 + 32 \pi ) square units
- Approximate: Total Area ( \approx 152.5 ) square units

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