Area of a Circle Formula in Terms of a Sector

The area of a circle is commonly expressed using the formula $A = \pi r^2$, where $r$ is the radius of the circle. However, in certain cases, we may need to find the area of a specific sector of a circle rather than the entire area. A sector of a circle is essentially a “slice” or a portion of the circle, bounded by two radii and the arc between them.

This article will explore how to derive the formula for the area of a sector in terms of the entire circle's area.

1. What is a Sector?

A sector of a circle is the region enclosed between two radii and the arc that connects their endpoints on the circumference of the circle. The central angle, $\theta$, of a sector is the angle formed by the two radii, measured in degrees or radians.

• A full circle has a central angle of $360^\circ$ (or $2\pi$ radians).
• A sector corresponds to a fraction of the full circle, depending on the central angle $\theta$.

2. Relationship Between Sector and Circle

The area of a sector is proportional to the central angle $\theta$ of the sector. To understand this, let’s first recall that the area of a full circle is given by the formula:

$$A = \pi r^2$$

Where:

• $A$ is the area of the circle,
• $r$ is the radius of the circle,
• $\pi$ is the mathematical constant approximately equal to 3.14159.

Since the area of the circle is proportional to the total angle of $360^\circ$ (or $2\pi$ radians), the area of a sector will be a fraction of the total area based on the central angle $\theta$. If the central angle is $\theta$, then the fraction of the circle's area that corresponds to the sector is $\frac{\theta}{360^\circ}$ (for degrees) or $\frac{\theta}{2\pi}$ (for radians).

3. Deriving the Formula for the Area of a Sector

Step 1: Expressing the Fraction of the Circle's Area

The area of the sector is a fraction of the entire circle's area. Using the central angle $\theta$, the area of the sector can be written as:

$$A_{\text{sector}} = \left( \frac{\theta}{360^\circ} \right) \times \pi r^2 \quad \text{(if \( \theta \) is in degrees)}$$

Or equivalently, if $\theta$ is measured in radians:

$$A_{\text{sector}} = \left( \frac{\theta}{2\pi} \right) \times \pi r^2 \quad \text{(if \( \theta \) is in radians)}$$

Step 2: Simplifying the Formula

By simplifying these expressions, we get the formulas for the area of a sector:

• For degrees:

  $$A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2$$

• For radians:

  $$A_{\text{sector}} = \frac{\theta}{2\pi} \times \pi r^2$$

Since the factor $\pi r^2$ represents the area of the entire circle, the formula for the area of the sector is a simple multiplication of the fraction of the circle (determined by the central angle) and the total area of the circle.

4. Examples

Example 1: Sector with Central Angle in Degrees

Suppose we have a circle with radius 5 cm and a sector with a central angle of $90^\circ$. To find the area of the sector:

• The total area of the circle is:

  $$A_{\text{circle}} = \pi (5^2) = 25\pi \approx. 78.54 \, \text{cm}^2$$

• The area of the sector is:

  $$A_{\text{sector}} = \frac{90^\circ}{360^\circ} \times 25\pi = \frac{1}{4} \times 25\pi = \frac{25\pi}{4} \approx. 19.63 \, \text{cm}^2$$

Example 2: Sector with Central Angle in Radians

Now, suppose the central angle of the sector is $\frac{\pi}{3}$ radians, and the radius of the circle is 6 cm. To calculate the area of the sector:

• The total area of the circle is:

  $$A_{\text{circle}} = \pi (6^2) = 36\pi \approx. 113.10 \, \text{cm}^2$$

• The area of the sector is:

  $$A_{\text{sector}} = \frac{\frac{\pi}{3}}{2\pi} \times 36\pi = \frac{1}{6} \times 36\pi = 6\pi \approx. 18.85 \, \text{cm}^2$$

The area of a sector is derived from the area of the entire circle and is directly proportional to the central angle of the sector. If the central angle is in degrees, the area of the sector is the fraction $\frac{\theta}{360^\circ}$ of the total area of the circle. If the central angle is in radians, the fraction is $\frac{\theta}{2\pi}$. Using these relationships, we can easily calculate the area of any sector, provided we know the radius of the circle and the central angle of the sector.