# 12.2 Circles

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This module introduces the definition and formula of a circle, including example problems.

## The definition of a circle

You’ve known all your life what a circle looks like. You probably know how to find the area and the circumference of a circle, given its radius. But what is the exact mathematical definition of a circle? Before you read the answer, you may want to think about the question for a minute. Try to think of a precise, specific definition of exactly what a circle is.

Below is the definition mathematicians use.

## Definition of a circle

The set of all points in a plane that are the same distance from a given point forms a circle . The point is known as the center of the circle , and the distance is known as the radius .

Mathematicians often seem to be deliberately obscuring things by creating complicated definitions for things you already understood anyway. But if you try to find a simpler definition of exactly what a circle is, you will be surprised at how difficult it is. Most people start with something like “a shape that is round all the way around.” That does describe a circle, but it also describes many other shapes, such as this pretzel:

So you start adding caveats like “it can’t cross itself” and “it can’t have any loose ends.” And then somebody draws an egg shape that fits all your criteria, and yet is still not a circle:

So you try to modify your definition further to exclude that ... and by that time, the mathematician’s definition is starting to look beautifully simple.

But does that original definition actually produce a circle? The following experiment is one of the best ways to convince yourself that it does.

## Experiment: drawing the perfect circle

1. Lay a piece of cardboard on the floor.
2. Thumbtack one end of a string to the cardboard.
3. Tie the other end of the string to your pen.
4. Pull the string as tight as you can, and then put the pen on the cardboard.
5. Pull the pen all the way around the thumbtack, keeping the string taut at all times.

The pen will touch every point on the cardboard that is exactly one string-length away from the thumbtack. And the resulting shape will be a circle. The cardboard is the plane in our definition, the thumbtack is the center , and the string length is the radius .

The purpose of this experiment is to convince yourself that if you take all the points in a plane that are a given distance from a given point, the result is a circle. We’ll come back to this definition shortly, to clarify it and to show how it connects to the mathematical formula for a circle.

## The mathematical formula for a circle

You already know the formula for a line: $y=mx+b$ . You know that $m$ is the slope, and $b$ is the y-intercept. Knowing all this, you can easily answer questions such as: “Draw the graph of $y=2x–3$ ” or “Find the equation of a line that contains the points (3,5) and (4,4).” If you are given the equation $3x+2y=6$ , you know how to graph it in two steps: first put it in the standard $y=mx+b$ form, and then graph it.

All the conic sections are graphed in a similar way. There is a standard form which is very easy to graph, once you understand what all the parts mean. If you are given an equation that is not in standard form, you put it into the standard form, and then graph it.

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
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Source:  OpenStax, Advanced algebra ii: conceptual explanations. OpenStax CNX. May 04, 2010 Download for free at http://cnx.org/content/col10624/1.15
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