Geometry Lesson 10.9: Arc Length

Introduction to Arc Length

• Arc length is different from arc measure; arc measure is in degrees, while arc length is the physical distance along the arc in units like centimeters or inches.
• This lesson (10.9) connects arc length to the broader topic of area covered in Chapter 11, although the concept originates from ideas discussed in Chapter 10 regarding circle geometry.

Central Angle and Arc Measure

• When discussing arcs, the central angle plays a significant role. In the given example:
  • Central angle (C) = 40 degrees.
  • This angle also represents the arc measure, so arc measure = 40 degrees.
• Visualize this scenario as someone trying to walk along the arc from point A to B in a circle.

Understanding Circumference

• The total distance around the circle is its circumference, calculated using the formula:
  • C = 2 \pi r
  • Given radius (r) = 8 cm, so:
  • C = 2 \pi \times 8 = 16\pi\ \, \text{cm}

Finding Arc Length

• To find the arc length of segment AB:
  • Determine what fraction of the whole circle the arc represents:
  • Arc measure (40 degrees) divided by full circle (360 degrees):
  • \frac{40}{360} = \frac{1}{9}
• Calculate the arc length using this fraction of the circumference:
  • Arc Length (L) = (Fraction of Circle) × Circumference
  • L = \frac{40}{360} \times 16\pi = \frac{1}{9} \times 16\pi = \frac{16\pi}{9}

Final Result

• The arc length of segment AB is:
  • \frac{16\pi}{9} \text{ cm}
• Important Note: When reporting answers involving pi, remember that pi is part of the number, not a unit label.

Key Formula

• The general formula for finding arc length:
  • [ \text{Arc Length} = \left( \frac{\text{Arc Measure (in degrees)}}{360} \right) \times \text{Circumference} \
  • This formula is your focus for practice problems and homework assignments.