Area of Sector Formula with Arc Length

Introduction

A sector is a portion of a circle bounded by two radii and the arc between them. While the area of a sector is typically calculated using the central angle, it can also be determined using the arc length. This method is particularly useful when the arc length is given instead of the angle.

This page explains how to derive the formula for the area of a sector using the arc length, provides detailed examples, and demonstrates how it simplifies certain calculations.

Derivation of the Formula

1. Understanding Arc Length

The arc length of a sector is the distance along the curved boundary of the sector. It is related to the central angle and the radius of the circle. The formula for arc length (\( L \)) is:

$$L=\theta \times r$$

Here:

• \( L \) = Arc length
• \( \theta \) = Central angle in radians
• \( r \) = Radius of the circle

2. Formula for the Area of a Sector Using Arc Length

The formula for the area of a sector using the central angle in radians is:

$$\mathrm{Area}=\frac{\theta r^2}{2}$$

Substituting \( \theta \) from the arc length formula \( \theta = \frac{L}{r} \):

$$\mathrm{Area}=\frac{\frac{L}{r}\times r^2}{2}$$

Simplifying:

$$\mathrm{Area}=\frac{L\times r}{2}$$

This is the formula for the area of a sector using the arc length.

Step-by-Step Example

Example 1: Simple Calculation

Let’s calculate the area of a sector with the following values:

• Arc length: $L=12 \mathrm{cm}$
• Radius: $r=4 \mathrm{cm}$

Step 1: Apply the formula:

$$\mathrm{Area}=\frac{L\times r}{2}$$

Step 2: Substitute the values:

$$\mathrm{Area}=\frac{12\times 4}{2}$$

Step 3: Simplify:

$$\mathrm{Area}=\frac{48}{2}=24 \mathrm{cm}{2}^{}$$

Answer: The area of the sector is 24 cm².

Conclusion

The formula for the area of a sector using arc length is:

$$\mathrm{Area}=\frac{L\times r}{2}$$

This method is particularly useful when the arc length is given directly instead of the central angle. Practice using this formula with varying values of \( L \) and \( r \) to gain confidence in solving geometry problems involving sectors.