11.6: Circles - Sectors and Segments

Area of a Circle

• The formula for the area of an entire circle is:
  A = ext{pi} imes r^2
  where $r$ is the radius of the circle.

Parts of a Circle

• Focus on two main parts of a circle:
  1. Sectors
  2. Segments

Sector of a Circle

• Definition: A sector is a region bounded by two radii and an arc of the circle.
• Analogy: Think of a sector as a slice of pizza.
• Formula for Area of a Sector:
  • Area of a sector = Area of the circle × (Arc Measure / 360 degrees)
  • For example:
  • Radius of circle = 12 inches
  • Arc measure = 45 degrees
  • Area of whole circle = ext{pi} imes (12)^2 = 144 ext{pi}
  • Fraction of circle = \frac{45}{360} = \frac{1}{8}
  • Area of the sector = \frac{1}{8} imes 144 ext{pi} = 18 ext{pi}
    • Final Answer: 18π inches².

Segment of a Circle

• Definition: A segment is a region bounded by a chord and its intercepted arc.
• Analogy: Think of a segment as the "crust" of the pizza slice.
• Formula for Area of a Segment:
  • Area of the segment = Area of the sector - Area of the triangle
  • For example:
  • Radius = 10 meters
  • Central angle = 90 degrees
  • Arc measure = 90 degrees
  • Area of sector = A{ ext{sector}} = A{ ext{circle}} \times \frac{90}{360}
    • A_{ ext{circle}} = 10^2 ext{pi} = 100 ext{pi}
    • A_{ ext{sector}} = 100 ext{pi} \times \frac{1}{4} = 25 ext{pi}
  • Area of the triangle:
    • Base = height = 10 (right triangle with both sides as radius)
    • Area = \frac{1}{2} \times 10 \times 10 = 50
  • Area of the segment = 25 ext{pi} - 50
    • Final Answer: (25π - 50) meters².

Conclusion

• Key formulas to remember:

  • Area of a sector = Area of circle × (Arc Measure / 360)
  • Area of segment = Area of sector - Area of triangle

• Understanding these concepts will enable you to analyze and solve problems related to sectors and segments of circles in geometry effectively.