Geometry (B): READ: Circumference and Arc Length Reading

Circumference and Arc Length

Learning Objectives

Understand the basic idea of a limit.
Calculate the circumference of a circle.
Calculate the length of an arc of a circle.

Introduction

In this lesson, we extend our knowledge of perimeter to the perimeter—or circumference—of a circle. We’ll use the idea of a limit to derive a well-known formula for the circumference. We’ll also use common sense to calculate the length of part of a circle, known as an arc.

The Parts of a Circle

A circle is the set of all points in a plane that are a given distance from another point called the center. Flat round things, like a bicycle tire, a plate, or a coin, remind us of a circle.

The diagram reviews the names for the “parts” of a circle.

The center
The circle: the points that are a given distance from the center (which does not include the center or interior)
The interior: all the points (including the center) that are inside the circle
circumference: the distance around a circle (exactly the same as perimeter)
radius: any segment from the center to a point on the circle (sometimes “radius” is used to mean the length of the segment and it is usually written as $r$ )
diameter: any segment from a point on the circle, through the center, to another point on the circle (sometimes “diameter” is used to mean the length of the segment and it is usually written as $d$ )

If you like formulas, you can already write one for a circle:

$d = 2r$ or $(r = \frac {d}{2})$

Circumference Formula

The formula for the circumference of a circle is a classic. It has been known, in rough form, for thousands of years. Let’s look at one way to derive this formula.

Start with a circle with a diameter of $1\;\mathrm{unit}$ . Inscribe a regular polygon in the circle. We’ll inscribe regular polygons with more and more sides and see what happens. For each inscribed regular polygon, the perimeter will be given (how to figure that is in a review question).

What do you notice?

The more sides there are, the closer the polygon is to the circle itself.
The perimeter of the inscribed polygon increases as the number of sides increases.
The more sides there are, the closer the perimeter of the polygon is to the circumference of the circle.

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 Diameter $1$

Number of sides of polygon	Perimeter of polygon
$3$	$2.598$
$4$	$2.828$
$5$	$2.939$
$6$	$3.000$
$8$	$3.062$
$10$	$3.090$
$20$	$3.129$
$50$	$3.140$
$100$	$3.141$
$500$	$3.141$

As the number of sides of the inscribed regular polygon increases, the perimeter seems to approach a “limit.” This limit, which is the circumference of the circle, is approximately $3.141$ . This is the famous and well-known number $\pi$ . $\pi$ is an endlessly non-repeating decimal number. We often use $\pi \approx 3.14$ as a value for $\pi$ in calculations, but this is only an approximation.

Conclusion: The circumference of a circle with diameter $1$ is $\pi$ .

For Further Reading

Mathematicians have calculated the value of $\pi$ to thousands, and even millions, of decimal places. You might enjoy finding some of these megadecimal numbers. Of course, all are approximately equal to $3.14$ .

The article at the following URL shows more than a million digits of the decimal for $\pi$ . http://wiki.answers.com/Q/What_is_the_exact_value_for_Pi_at_this_moment

Tech Note - Geometry Software

You can use geometry software to continue making more regular polygons inscribed in a circle with diameter $1$ and finding their perimeters.

Can we extend this idea to other circles? First, recall that all circles are similar to each other. (This is also true for all equilateral triangles, all squares, all regular pentagons, etc.)

Suppose a circle has a diameter of $d \;\mathrm{units}$ .

The scale factor of this circle and the one in the diagram and table above, with diameter $1$ , is $d: 1,$ , $\frac{d}{1}$ , or just $d$ .
You know how a scale factor affects linear measures, which include perimeter and circumference. If the scale factor is $d$ , then the perimeter is $d$ times as much.

This means that if the circumference of a circle with diameter $1$ is $\pi$ , then the circumference of a circle with diameter $d$ is $\pi d$ .

Circumference Formula

Let $d$ be the diameter of a circle, and $C$ the circumference.

$C = \pi d$

Example 1

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

Use $C = \pi d$ . The length of a side of the square is also the diameter of the circle. $C = \pi d = 10\pi \approx 31.4 \;\text{ cm}$

Note that sometimes an approximation is given using $\pi \approx 3.14$ . In this example the circumference is $31.4\;\mathrm{cm}$ using that approximation. An exact is given in terms of $\pi$ (leaving the symbol for $\pi$ in the answer rather than multiplying it out. In this example the exact circumference is $10 \pi\;\mathrm{cm}$ .

Arc Length

Arcs are measured in two different ways.

Degree measure: The degree measure of an arc is the fractional part of a $360^\circ$ complete circle that the arc is.
Linear measure: This is the length, in units such as centimeters and feet, if you traveled from one end of the arc to the other end.

Example 2

Find the length of $\widehat{PQ}$ .

$m\widehat{PQ}$ = $60^\circ$ . The radius of the circle is $9\;\mathrm{inches}$ .

Remember, $60^\circ$ is the measure of the central angle associated with $m\widehat{PQ}$ .

$m\widehat{PQ}$ is $\frac{60}{360}$ of a circle. The circumference of the circle is

$\pi d = 2\pi r = 2\pi(9) = 18\pi\;\mathrm{inches}$ . The arc length of $PQ:$ is $\frac{60}{360}\times 18\pi=\frac {1}{6}\times 18\pi=3 \pi \approx 9.42\;\mathrm{inches}$ .

In this lesson we study the second type of arc measure—the measure of an arc’s length. Arc length is directly related to the degree measure of an arc.

Suppose a circle has:

circumference $C$
diameter $d$
radius $r$

Also, suppose an arc of the circle has degree measure $m$ .

Note that $\frac{m}{360}$ is the fractional part of the circle that the arc represents.

Arc length

$\text{Arc Length}=\frac{m}{360}\times c=\frac {m}{360}\times \pi d=\frac {m}{360}\times 2\pi r$

Lesson Summary

This lesson can be summarized with a list of the formulas developed.

Radius and diameter: $d = 2 r$
Circumference of a circle: $C = \pi d$
$\mathrm{Arc\ length} = \frac{m}{360}\times c=\frac {m}{360}\times \pi d=\frac {m}{360}\times 2\pi r$

Points to Consider

After perimeter and circumference, the next logical measure to study is area. In this lesson, we learned about the perimeter of a circle (circumference) and the arc length of a sector. In the next lesson we’ll learn about the areas of circles and sectors.

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

Review Questions

Prove: The circumference of a circle with radius $r$ is $2\pi r.$
The Olympics symbol is five congruent circles arranged as shown below. Assume the top three circles are tangent to each other.

Brad is tracing the entire symbol for a poster. How far will his pen point travel?

A truck has tires that measure from the center of the wheel to the outer edge of the tire.
1. How far forward does the truck travel every time a tire turns exactly once?
2. How many times will the tire turn when the truck travels $1\;\mathrm{mile}$ ? $(1\;\mathrm{mile} = 5280\;\mathrm{feet})$ .
The following wire sculpture was made from two perpendicular segments that intersect each other at the center of a circle.
1. If the radius of the circle is $25\;\mathrm{cm}$ , how much wire was used to outline the shaded sections?
The circumference of a circle is $300\;\mathrm{feet}$ . What is the radius of the circle?
A gear with a radius of $3\;\mathrm{inches}$ inches turns at a rate of $2000$ RPM (revolutions per minute). How far does a point on the edge of the pulley travel in one second?
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. How far does the tip of the boom travel in one day?
The radius of Earth at the Equator is about . Belem (in Brazil) and the Galapagos Islands (in the Pacific Ocean) are on (or very near) the Equator. The approximate longitudes are Belem, , and Galapagos Islands, .
1. What is the degree measure of the major arc on the Equator from Belem to the Galapagos Islands?
2. What is the distance from Belem to the Galapagos Islands on the Equator the “long way around?”
A regular polygon inscribed in a circle with diameter $1$ has $n$ sides. Write a formula that expresses the perimeter, $p$ , of the polygon in terms of $n$ . (Hint: Use trigonometry.)
The pulley shown below revolves at a rate of RPM.
1. How far does point $A$ travel in one hour?