Calculating the Area of a Sector and Length of an Arc
Key Concepts
- Sector and Arc: A circle can be divided into sectors and arcs:
- Minor Sector: Smaller part of the circle with its corresponding minor arc and two radii.
- Major Sector: Larger part of the circle with its corresponding major arc and two radii.
- Angle and Radius: Questions often involve the angle of the minor sector and the radius of the circle.
Formulas
- Area of Sector: ( \text{Area} = \frac{x}{360} \times \pi r^2 )
- Length of Arc: ( \text{Length} = \frac{x}{360} \times 2\pi r )
- x is the angle of the sector in degrees.
- r is the radius of the circle.
Conceptual Understanding
- Think of the sector or arc as a fraction of the entire circle:
- Example: If ( x = 90^\circ ), the sector is one-quarter of the circle (since 90 is one-quarter of 360).
- Area: ( \frac{1}{4} \times \pi r^2 )
- Arc Length: ( \frac{1}{4} \times 2\pi r )
Examples
-
Simple Angles:
- Angle: 90 degrees
- Radius: 6 cm
- Area: ( \frac{1}{4} \times \pi \times 6^2 = 28.3 \text{ cm}^2 )
- Arc Length: ( \frac{1}{4} \times 2 \times \pi \times 6 = 9.42 \text{ cm} )
-
Complex Angles:
- Angle: 113 degrees
- Use fractions with denominator 360.
- Radius: 15 mm
- Area: ( \frac{113}{360} \times \pi \times 15^2 = 222 \text{ mm}^2 )
- Arc Length: ( \frac{113}{360} \times 2 \times \pi \times 15 = 29.6 \text{ mm} )
Practice Example
- Sector OAB:
- Angle: 70 degrees
- Radius: 14 cm
- Area: ( \frac{70}{360} \times \pi \times 14^2 = 120 \text{ cm}^2 )
- Arc Length: ( \frac{70}{360} \times 2 \times \pi \times 14 = 17.1 \text{ cm} )
Additional Resources
- Practice questions are available on the revision site linked in the video.
This video explains the calculations for sectors and arcs in a circle and provides a practical approach to understanding the underlying concepts of these mathematical formulas.