Area of Sector Formula (Radians)

Introduction

A sector is a portion of a circle formed by two radii and the arc between them. Calculating the area of a sector is essential in solving geometric problems. While angles are often expressed in degrees, using radians simplifies many mathematical and physical calculations.

This page explains how to calculate the area of a sector when the central angle is given in radians, provides a detailed derivation of the formula, and compares it with the degree-based formula. To learn about the degree-based formula, visit the Area of Sector Formula (Degrees) page.

Derivation of the Formula

1. Understanding Radians

A radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. Since the circumference of a circle is \( 2\pi r \), a full circle measures \( 2\pi \) radians.

2. Formula for the Area of a Sector

The area of a full circle is:

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

A sector represents a fraction of the circle. When the central angle \( \theta \) is given in radians, the fraction of the circle covered by the sector is:

$$\frac{\theta }{\mathrm{2\pi }}$$

Multiplying this fraction by the total area of the circle gives the area of the sector:

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

Simplifying this expression:

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

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

Step-by-Step Example

Example 1: Simple Calculation

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

• Central angle: $\theta =\frac{\pi }{3}$ radians
• Radius: $r=6 \mathrm{cm}$

Step 1: Apply the formula:

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

Step 2: Substitute the values:

$$\mathrm{Area}=\frac{\frac{}{}}{}$$