A sector is a portion of a circle, resembling a ‘slice of pie.’ It is defined by two radii and the arc between them.
Key Components of a Sector
Radius ($r$)
The radius is the distance from the center of the circle to any point on its circumference. In the context of a sector, the radius remains constant.
Central Angle ($theta$)
The central angle is the angle formed at the center of the circle by the two radii. It is typically measured in degrees or radians.
Formula for the Area of a Sector
The formula for the area of a sector depends on the central angle and the radius of the circle. Here’s how you can calculate it:
When the Angle is in Degrees
If the central angle ($theta$) is given in degrees, the formula is:
$A = frac{theta}{360} times pi r^2$
When the Angle is in Radians
If the central angle ($theta$) is given in radians, the formula simplifies to:
$A = frac{1}{2} theta r^2$
Example Problems
Example 1: Angle in Degrees
Suppose you have a sector with a radius of 5 cm and a central angle of 60 degrees. To find the area, use the formula:
$A = frac{60}{360} times pi times 5^2$
Simplify it step by step:
$A = frac{1}{6} times pi times 25$
$A = frac{25pi}{6} approx 13.09 text{ cm}^2$
Example 2: Angle in Radians
Now, let’s consider a sector with a radius of 4 meters and a central angle of $frac{pi}{3}$ radians. The formula becomes:
$A = frac{1}{2} times frac{pi}{3} times 4^2$
Simplify it step by step:
$A = frac{1}{2} times frac{pi}{3} times 16$
$A = frac{8pi}{3} approx 8.38 text{ m}^2$
Conclusion
Understanding the formula for the area of a sector allows you to solve real-world problems involving circular segments. Whether you’re working with degrees or radians, the key is to remember the relationship between the central angle and the radius.