Area of Sector Formula (Degrees)

Introduction

A sector is a portion of a circle enclosed by two radii and the arc between them. In geometry, calculating the area of a sector is a fundamental concept, often applied in solving problems involving circular regions.

When the central angle of the sector is measured in degrees, the area of the sector can be determined using a specific formula. This page will explain the derivation of the formula, provide example calculations, and compare it with the formula for sectors when the angle is measured in radians.

Derivation of the Formula

To derive the formula for the area of a sector, let’s start by understanding the concept of a fraction of a circle.

1. Fraction of the Circle

A full circle has an area given by:

$$\mathrm{Area}=\pi r^2$$

The circle is divided into 360 degrees. The fraction of the circle occupied by a sector with a central angle θ (in degrees) is:

$$\frac{\theta }{360}$$

2. Area of the Sector

Since the area of a full circle is $\pi r^2$, the area of the sector can be found by multiplying the fraction of the circle by the total area:

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

This is the formula for the area of a sector when the central angle is given in degrees.

Example Calculations

Example 1: Simple Numbers

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

• Central angle, θ = 60°
• Radius, r = 5 cm

Step 1: Calculate the fraction of the circle:

$$\frac{60}{360}=\frac{1}{6}$$

Step 2: Compute the total area of the circle:

$$\pi \times r^2=25\pi \text{cm²}$$

Step 3: Multiply the fraction by the circle’s area:

$$\frac{25}{6}\pi \text{cm²}$$

Answer: The area of the sector is approximately 13.09 cm².

Example 2: Slightly Complex Calculation

For θ = 45° and r = 7 cm:

Step 1: Fraction of the circle:

$$\frac{45}{360}=\frac{1}{8}$$

Step 2: Total area of the circle:

$$49\pi \text{cm²}$$

Step 3: Area of the sector:

$$\frac{49}{8}\pi \text{cm²}$$

Answer: The area of the sector is approximately 19.25 cm².

Comparison to Radians Formula

If the angle is given in radians instead of degrees, the formula changes to:

$$\frac{\theta }{2}{r}^2$$

Learn more about the Area of Sector Formula (Radians).

Conclusion

The formula for the area of a sector in degrees is:

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

Always verify whether the angle is in degrees or radians before using the formula. Practicing with problems involving varying radii and angles will help you gain confidence in using this formula.