6.3 Circumference & Area of Sectors

Learning Objectives

You will demonstrate an understanding of calculating the circumference of a circle.
You will be able to find the area of a sector given information such as the circumference and area of a sector.
You will be able to calculate the area of a sector given the radius and the central angle measure.
You will be able to understand what is meant by a sector of a circle.
You will be able to find the area of sectors given radius and central angle measure.
You will be able to find the arc length of a sector given radius and central angle measure.

Introduction

What if you were given a circle with two radii in which the region between those two radii was shaded? How could you find the area of that shaded region of the circle? Can you calculate the length of the arc?

This lesson will help you learn how to find the measurements of a shaded sector, including its area and the arc length.

Guided Learning

Find the area of a sector

A sector of a circle is the area bounded by two radii and the arc between the endpoints of the radii. The sector is like a slice of the circle, so its area will be a portion of the whole thing. To figure out how big the "slice" is, first figure out the size of the whole circle. You learned how to do this already, remember it is $A= \pi r^2$ . Now we want to take only part of this circle, so we figure out how the angle measure of the slice compares to the entire circle. Try to look at this value as a fraction of the whole thing. We now can calculate the sector's area by multiplying these two values in the formula $A=\frac{m \widehat{AB}}{360^\circ} \cdot \pi r^2$ If is the radius and $\widehat{AB}$ is the arc bounding a sector.

Each of these shows what a sector may look like.

Let’s look at an example.

Find the area of the blue sector. Leave your answer in terms of $\pi$ .

In the picture, the central angle that corresponds with the sector is $60^\circ$ . $60^\circ$ is $\frac{1}{6}$ of $360^\circ$ , so this sector is $\frac{1}{6}$ of the total area. $area \ of \ blue \ sector=\frac{1}{6} \cdot \pi 8^2=\frac{32}{3} \pi$ This answer was left in terms of $\pi$ , but could also be approximated by using the value 3.14. In that case $\frac{1}{6}\cdot 3.14\cdot8^2 = \frac{1}{6}\cdot 3.14\cdot64 = \frac{1}{6} \cdot \ 200.96 \approx 33.49$ .

Now, give it a try.

Find the area of the blue sector of $\bigodot A$ .

Take a few minutes to check your work.

Now that you understand how to calculate the area of a sector, let's apply a similar strategy to find the arc length of the same sector. This time we need to recall the other formula that we learned in relation to circles. Remember the formula for circumference is $C=2\pi r$ . We'll use the formula given radius, since the sector will be created with radii. Since we only want a "slice" of the whole circle, we only want a small section of the total circumference. Again, we need to figure out the fractional part of the whole circle, but dividing the central angle measure by $360^\circ$ . Multiply this fraction by the circumference and there you have it! Your arc length of the sector. The formula looks like $Arc Length = \frac{m\widehat{AB}}{360^\circ } \cdot 2\pi r$ If is the radius and $\widehat{AB}$ is the arc bounding a sector.

Let’s look at an example.

Find the arc length of the blue sector. Leave your answer in terms of $\pi$ .

In the picture, the central angle that corresponds with the sector is $60^\circ$ . $60^\circ$ is $\frac{1}{6}$ of $360^\circ$ , so this sector is $\frac{1}{6}$ of the total area. $Arc \ length \ of \ blue \ sector = \frac{1}{6} \cdot 2\pi \left ( 8 \right ) = \frac{8}{3} \pi$ This answer was left in terms of $\pi$ , but could also be approximated by using the value 3.14. In that case $\frac{1}{6}\cdot 2 \left ( 3.14 \right ) 8 =\frac{1}{6}\cdot \left ( 6.28 \right ) 8 = \frac{1}{6}\cdot \left ( 50.24 \right ) \approx 8.37$ .