Calculating the Area of a Sector and the Length of an Arc

Key Concepts

• Sector: A "slice" of a circle, defined by two radii and the arc between them.
  • Minor Sector: Smaller slice with a smaller arc.
  • Major Sector: Larger slice with a larger arc.
• Arc: The curve between two points on the circle.
• Angle (x): Usually given in problems alongside the radius.

Formulas

• Area of a Sector: ( \frac{x}{360} \times \pi r^2 )
• Length of an Arc: ( \frac{x}{360} \times 2\pi r )

Understanding the Formulas

• Conceptualize the sector/arc as a fraction of the entire circle.
• A circle has 360 degrees.
• Example: 90 degrees is 1/4 of a circle; thus, the area is 1/4 of the circle's area and the arc length is 1/4 of the circumference.

Steps with Examples

1. For a 90-degree sector with radius 6 cm:
   • Area: ( \frac{1}{4} \times \pi \times 6^2 = 28.3 \text{ sq cm} )
   • Arc Length: ( \frac{1}{4} \times 2 \times \pi \times 6 = 9.42 \text{ cm} )
2. For a 113-degree sector with radius 15 mm:
   • Area: ( \frac{113}{360} \times \pi \times 15^2 = 222 \text{ sq mm} )
   • Arc Length: ( \frac{113}{360} \times 2 \times \pi \times 15 = 29.6 \text{ mm} )

Application

• In exam questions, identify the angle and radius and use the formulas.
• Use fractions with a denominator of 360 for non-standard angles.

Practice Problem

• Given: Sector OAB, angle 70 degrees, radius 14 cm.
  • Area of Sector OAB:
    • Calculation: ( \frac{70}{360} \times \pi \times 14^2 = 120 \text{ sq cm} )
  • Length of Arc AB:
    • Calculation: ( \frac{70}{360} \times 2 \times \pi \times 14 = 17.1 \text{ cm} )

Additional Resources

• Practice problems available on the revision site (link provided in the video).