Lecture Notes: Arc Length and Area of Sectors

Key Topics

• Understanding arc length
• Calculating arc length in radians and degrees
• Area of a sector
• Converting between degrees and radians

Arc Length

Definition

• Arc Length (s): The distance along the curved line making up the arc.
  • Notation: Often represented as 's'.
  • Based on the angle ( \theta ) and the radius ( r ).

Formula for Arc Length

• For angle in radians: [ s = \theta \times r ]

  • Example: Angle of 5 radians and radius of 12T
    • ( s = 5 \times 12 = 60 ) units

• For angle in degrees: [ s = \frac{\theta}{360} \times 2\pi r ]

  • Example: ( \theta = 150^\circ ), ( r = 8 \text{ cm} )
    • Convert degrees to radians: ( \theta = \frac{150 \times \pi}{180} = \frac{5\pi}{6} )
    • Calculate arc length: ( s = \frac{5\pi}{6} \times 8 = \frac{40\pi}{6} = \frac{20\pi}{3} )
    • Decimal approximation: ( \approx 20.944 \text{ cm} )

Conversion Between Degrees and Radians

• [ \text{Radians} = \frac{\pi}{180} \times \text{Degrees} ]

Area of a Sector

Formula for Area

• For angle in radians: [ A = \frac{1}{2} \theta r^2 ]

• For angle in degrees: [ A = \frac{\theta}{360} \times \pi r^2 ]

Examples

• Angle in degrees: ( \theta = 90^\circ, r = 10 \text{ cm} )

  • ( A = \frac{90}{360} \times \pi \times 10^2 = 25\pi ) square cm

• Angle in radians: ( \theta = 2, r = 8 \text{ cm} )

  • ( A = \frac{1}{2} \times 2 \times 8^2 = 64 ) square cm

Additional Resources

• Online Course: For further study, visit emi.com and search for trigonometry.
  • Course includes:
    • Introduction to angles
    • Converting degrees to radians
    • Arc Length, Unit Circle, Trigonometric Functions
    • Graphing functions, solving problems

Note

• Remember these two key formulas for both arc length and area of a sector.
• Practice converting between degrees and radians for accuracy in calculations.