Calculating the Area of Shaded Regions

Introduction

• Calculate the area of shaded regions by finding the difference between the area of a larger object and a smaller object within it.

Example 1: Square Inside a Rectangle

• Square side length = 5 inches
• Rectangle dimensions = 8 inches by 10 inches
• Area Calculation:
  • Area of Rectangle = length × width = 10 × 8 = 80 square inches
  • Area of Square = side² = 5² = 25 square inches
  • Area of Shaded Region = Area of Rectangle - Area of Square = 80 - 25 = 55 square inches

Example 2: Circle Inside a Square

• Circle radius = 8 units
• Square side length = 20 units
• Area Calculation:
  • Area of Square = side² = 20² = 400 square units
  • Area of Circle = π × radius² = π × 8² = 64π square units
  • Area of Shaded Region = 400 - 64π
  • Decimal approximation = 198.9 square units

Example 3: Circle Within Another Circle

• Inner Circle radius = 4 units
• Outer Circle radius = 7 units
• Area Calculation:
  • Area of Large Circle = π × 7² = 49π
  • Area of Small Circle = π × 4² = 16π
  • Area of Shaded Region = 49π - 16π = 33π

Example 4: Circle Inscribed in a Triangle

• Circle Radius = 8
• Area Calculation:
  • Large Circle Area = π × radius² = π × 8² = 64π
  • Triangle Area (right angle) = 1/2 × base × height = 1/2 × 8 × 8 = 32
  • Area of Shaded Region = 64π - 32
  • Decimal approximation = 169.1 square units

Example 5: Rhombus Inside a Rectangle

• Rectangle width = 8, length = calculated using rhombus properties
• Rhombus Diagonals meet at right angles
• Area Calculation:
  • Rectangle Area = length × width
  • Rhombus Area = 1/2 × diagonal1 × diagonal2
  • If D1 = 8, then diagonals = 4 and 4, form right triangles
  • Length = 3 + 3 = 6 (using Pythagorean theorem)
  • Area of Shaded Region = (1/2 × 6 × 8) = 24 square units

Example 6: Equilateral Triangle Around a Circle

• Circle Radius = 20
• Equilateral Triangle formed, using properties of inscribed circle
• Area Calculation:
  • Triangle Area = (√3/4) × side²
  • Side of Triangle = 40√3
  • Circle Area = π × radius² = 400π
  • Area of Shaded Region = 1200√3 - 400π

Conclusion

• The method involves calculating the area of the larger shape and subtracting the area of the smaller shape to find the area of the shaded region.
• Different shapes and configurations require using specific geometric formulas.