Circle Measurements and Graphs: Sectors

In this lesson, we will explore sectors of circles, which are portions of a circle defined by two radii and the arc between them. The area of a sector can be calculated based on the central angle and the radius of the circle.

Area of a Sector

The area $A$ of a sector can be calculated using the formula:

$$A = \frac{\theta}{360} \times \pi r^{2}$$

where:

• $\theta$ is the central angle in degrees
• $r$ is the radius of the circle

Worked Example

Find the area of a sector with a radius of 5 units and a central angle of 60 degrees.

1. Identify the values:
   • Radius $r = 5$
   • Central angle $\theta = 60$
2. Substitute into the area formula: $$A = \frac{60}{360} \times \pi (5^{2})$$
3. Simplify the fraction: $$A = \frac{1}{6} \times \pi (25)$$
4. Calculate the area: $$A = \frac{25\pi}{6}$$

Thus, the area of the sector is $\frac{25\pi}{6}$ square units.

Key Questions

• What is the area of a sector with a radius of 10 units and a central angle of 90 degrees?
• How does the area change if the radius is doubled?
• If the area of a sector is $20\pi$ square units and the radius is 8 units, what is the central angle?
• Calculate the area of a sector where the central angle is 180 degrees and the radius is 4 units.

Keywords

• Sector
• Area
• Radius
• Central angle
• Circle