7th Grade Math > Circles and Sectors > Area and Arc Length of a Sector, Formulas and Principles, Practice Worksheets

In the unit on circles and sectors, there are many formulas for the perimeter and area of sectors. Understanding the sector's perimeter and area as extensions of the circle's perimeter and area can help you easily grasp and memorize these formulas.

First, what is a sector?

Simply put, a sector is a portion of a circle. More specifically, it includes a part of the circle that contains the center. The diagram below helps illustrate this concept.

Calculating the area and arc length of a sector is closely related to finding the perimeter and area of a circle. The concept of ratios plays a key role in understanding sectors.

Ratio

• Ratio: A comparison of two quantities.
• Compared Quantity: The part being measured.
• Base Quantity: The reference amount.

For sectors, the base quantity is the angle of 360 degrees.

How do we find the arc length of a sector? When you look at the diagram, it becomes quite intuitive.

If the circumference of a circle is $C$, what is the arc length of the orange-colored sector? It's calculated based on the proportion of the sector to the entire circle.

Thus, the arc length of a sector is a part of the circle's circumference proportional to the angle of the sector. Let's turn this into a formula.

In mathematics, the angle of 360 degrees is often expressed as $2\pi$ radians. Although using radians might seem confusing at first, with practice, it becomes more familiar.

If the angle of the sector is $\theta$ radians, the formula for the arc length of the sector is:

$\text{Arc Length of Sector} = \frac{\theta}{2\pi} \times \text{Circumference of Circle}$

How about the area of a sector? Similarly, we consider the area of the circle and the proportion of the sector.

Since a sector is a part of a circle, we start by calculating the area of the circle and then apply the proportion of the sector.

In the diagram, if the area of the circle is $A$ and the proportion of the sector to the circle is $\frac{\theta}{2\pi}$, the area of the sector is:

$\text{Area of Sector} = \frac{\theta}{2\pi} \times \text{Area of Circle}$

Understanding these principles leads us to the following formulas:

$\text{Arc Length of Sector} = \frac{\theta}{2\pi} \times 2\pi r = \theta r$

$\text{Area of Sector} = \frac{\theta}{2\pi} \times \pi r^2 = \frac{1}{2} \theta r^2$

Summary

We studied the formulas and principles for the perimeter and area of a sector.

For practice, visit the 'ModuMath' website, where you can generate custom worksheets to reinforce your learning.

Click the image below to go to Modoo Math.