Geometry (B): READ: Circles and Sectors Reading

Circles and Sectors

Learning Objectives

Calculate the area of a circle.
Calculate the area of a sector.
Expand understanding of the limit concept.

Introduction

In this lesson we complete our area toolbox with formulas for the areas of circles and sectors. We’ll start with areas of regular polygons, and work our way to the limit, which is the area of a circle. This may sound familiar; it’s exactly the same approach we used to develop the formula for the circumference of a circle.

Area of a Circle

The big idea:

Find the areas of regular polygons with radius $1$ .
Let the polygons have more and more sides.
See if a limit shows up in the data.
Use similarity to generalize the results.

The details:

Begin with polygons having $3, 4,$ and $5$ sides, inscribed in a circle with a radius of $1$ .

Now imagine that we continued inscribing polygons with more and more sides. It would become nearly impossible to tell the polygon from the circle. The table below shows the results if we did this.

Regular Polygons Inscribed in a Circle with Radius $1$

Number of sides of polygon	Area of polygon
$3$	$1.2990$
$4$	$2.0000$
$5$	$2.3776$
$6$	$2.5981$
$8$	$2.8284$
$10$	$2.9389$
$20$	$3.0902$
$50$	$3.1333$
$100$	$3.1395$
$500$	$3.1415$
$1000$	$3.1416$
$2000$	$3.1416$

As the number of sides of the inscribed regular polygon increases, the area seems to approach a “limit.” This limit is approximately $3.1416$ , which is $\pi$ .

Conclusion: The area of a circle with radius $1$ is $\pi$ .

Now we extend this idea to other circles. You know that all circles are similar to each other.

Suppose a circle has a radius of $r \;\mathrm{units}$ .

The scale factor of this circle and the one in the diagram and table above, with radius $1$ , is $r: 1, \frac{r}{1}$ , or just $r$ .
You know how a scale factor affects area measures. If the scale factor is $r$ , then the area is $r^2$ times as much.

This means that if the area of a circle with radius $1$ is $\pi$ , then the area of a circle with radius $r$ is $\pi r^2$ .

Area of a Circle Formula

Let $r$ be the radius of a circle, and $A$ the area.

$A=\pi r^2$

You probably noticed that the reasoning about area here is very similar to the reasoning in an earlier lesson when we explored the perimeter of polygons and the circumference of circles.

Example 1

A circle is inscribed in a square. Each side of the square is $10\;\mathrm{cm}$ long. What is the area of the circle?

Use $A=\pi r^2$ . The length of a side of the square is also the diameter of the circle. The radius is $5 \;\mathrm{cm}$ .

$A=\pi {r^2} =\pi(5^2)= 25\pi \approx {78.5}$

The area is $25\pi\thickapprox 78.5 \;\mathrm{cm}^2$ .

Area of a Sector

The area of a sector is simply an appropriate fractional part of the area of the circle. Suppose a sector of a circle with radius $r$ and circumference $C$ has an arc with a degree measure of $m^\circ$ and an arc length of $s\;\mathrm{units}$ .

The sector is $\frac{m}{360}$ of the circle.
The sector is also $\frac{s}{c} = \frac{s}{{2}{\pi}{r}}$ of the circle.

To find the area of the sector, just find one of these fractional parts of the area of the circle. We know that the area of the circle is $\pi r^2$ . Let $A$ be the area of the sector.

$A = \frac{m}{360}\times \pi r^2$

Also, $A = \frac{s}{c}\times\pi r^2 = \frac{s}{{2}{\pi}{r}}\times\pi r^2 =\frac{1}{2}sr.$

Area of a Sector

A circle has radius $r$ . A sector of the circle has an arc with degree measure $m^\circ$ and arc length $s\;\mathrm{units}$ .

The area of the sector is $A \;\mathrm{square\ units}$ .

$A = \frac {m}{360}\times \pi r^2 = \frac{1}{2}sr$

Example 2

Mark drew a sheet metal pattern made up of a circle with a sector cut out. The pattern is made from an arc of a circle and two perpendicular $6-\mathrm{inch}$ radii.

How much sheet metal does Mark need for the pattern?

The measure of the arc of the piece is $270^\circ$ , which is $\frac{270}{360}=\frac {3}{4}$ of the circle.

The area of the sector (pattern) is $= \frac{3}{4}\pi r^2=\frac {3}{4}\pi \times 6^2 = 27\pi \thickapprox 84.8\;\mathrm{sq\ in}$ .

Lesson Summary

We used the idea of a limit again in this lesson. That enabled us to find the area of a circle by studying polygons with more and more sides. Our approach was very similar to the one used earlier for the circumference of a circle. Once the area formula was developed, the area of a sector was a simple matter of taking the proper fractional part of the whole circle.

Summary of Formulas:

Area Formula

Let $r$ be the radius of a circle, and $A$ the area.

$A = \pi r^2$

Area of a Sector

A circle has radius $r$ . A sector of the circle has an arc with degree measure $m^\circ$ and arc length $s\;\mathrm{units}$ .

The area of the sector is $A$ square units.

$A=\frac {m}{360}\times \pi r^2 =\frac{1}{2}sr$

Points to Consider

When we talk about a limit, for example finding the limit of the areas of regular polygons, how many sides do we mean when we talk about “more and more?” As the polygons have more and more sides, what happens to the length of each side? Is a circle a polygon with an infinite number of sides? And is each “side” of a circle infinitely small? Now that’s small!

In the next lesson you’ll see where the formula comes from that gives us the areas of regular polygons. This is the formula that was used to produce the table of areas in this lesson.

The following questions are for your own review. The answers below are for you to check your work and understanding.

Review Questions

Complete the table of radii and areas of circles. Express your answers in terms of $\pi$ .

	Radius (units)	Area (square units)
1a.	$10$	?
1b.	?	$2.25 \pi$
1c.	?	$9$
1d.	$5 \pi$	?

Prove: The area of a circle with diameter $d$ is $\frac{\pi d^2}{4}.$
A circle is inscribed in a square.

The yellow shaded area is what percent of the square?

The circumference of a circle is $300\;\mathrm{feet}$ . What is the area of the circle?
A center pivot irrigation system has a boom that is $400 \;\mathrm{m}$ long. The boom is anchored at the center pivot. It revolves around the center pivot point once every three days, irrigating the ground as it turns. How many hectares of land are irrigated each day?
$(1\;\mathrm{hectare} = 10,000 \;\mathrm{m}^2)$
Vicki is cutting out a gasket in her machine shop. She made a large circle of gasket material, then cut out and removed the two small circles. The centers of the small circles are on a diameter of the large circle. Each square of the grid is $1 \;\mathrm{square\ inch}$ .

How much gasket material will she use for the gasket?

A security system scans all points up to $100 \;\mathrm{m}$ from is base. It scans back and forth through an angle of $65^\circ$ .

How much space does the system cover?

A simplified version of the international radiation symbol is shown below.

The symbol is made from two circles and three equally spaced diameters of the large circle. The diameter of the large circle is $12\;\mathrm{inches}$ , and the diameter of the small circle is $4\;\mathrm{inches}$ . What is the total area of the symbol?

Chad has $400\;\mathrm{feet}$ of fencing. He will use it all. Which would enclose the most space, a square fence or a circular fence? Explain your answer.

Review Answers

1a. $100 \pi$ 1b. $1.5$ 1c. $\frac{3 \sqrt{\pi}}{\pi}$ 1d. $\left(25 \pi\right)^2$
$A& = \pi r^2, r = \frac{d}{2}\\ A & =\pi \left( \frac{d}{2}\right)^2=\pi \left ( \frac{d^2}{4} \right )=\frac{\pi d^2}{4}$
Approximately $21.5\%$
Approximately $7166\;\mathrm{square\ feet}$
Approximately $16.7$
Approximately $87.9\;\mathrm{square\ inches}$
Approximately $5669 \;\mathrm{m}^2$
$20 \pi \approx 62.8\;\mathrm{square\ inches}$
The circular fence has a greater area.
Square:

$P & = 4S = 400, s = 100\\ A & = s^2 = 100^2 = 10,000 \;\mathrm{ft}^2$

Circle:

$C & = \pi \ d = 2\pi r = 400\\ r & =\frac{400}{2 \pi}\\ A & ={\pi}r^2=\pi\left(\frac{400}{{2}{\pi}}\right)^2= \frac{{200}^2}{\pi}\thickapprox 12,739 \;\mathrm{ft}^2$

Last modified: Tuesday, June 29, 2010, 12:00 PM