Why Are There 2Pi Radians in a Circle?

Date: 11/24/2003 at 14:57:58
From: peter radian
Subject: 2 PI RADIANS

Why are there two pi radians in a circle?  I know that it has
something to do with the formula for the circumference of the circle,
but I'm not sure how it works.

Date: 11/24/2003 at 16:10:16
From: Doctor Schwa
Subject: Re: 2 PI RADIANS

Hi Peter,

The first thing you need to think about is how we define an angle of 
one radian.  Suppose you have a piece of string that is used as the 
radius of a circle: hold one end of it still, and attach a pen to the 
other end to trace the circle.

Now, how long would the distance around the circle, or the
circumference, be?

Let's measure it using the piece of string as your measuring tool. 
Lay the string on the circle.  One piece of string long would make an
arc of the circle equal to one radius in length.  Now draw a radius of
the circle where the string begins and another where it ends.  The
angle formed between those two radii is one radian.  If you've done 
this right, that angle should look to be roughly 60 degrees.

Now move your string so it starts where the last one ended and draw 
another angle of one radian.  If you continue to do this all the way 
around the circle, you will find that after 6 radians you are almost 
all the way around.  This suggests that the number of radians in a
full circle is a little more than 6.

How much more than 6 is it?  Well, let's think about the forumla for 
the circumference of a circle, which is pi times diameter or 2 times
pi times radius:

C = 2 * pi * r

Since the radius is the same as the length of string you are using, we 
can think of this as:

C = 2pi * string length

We're trying to figure out how many of those string lengths we need to 
go all the way around the circle, so let's divide the circumference by 
the string length and find out.  Using the last equation and dividing 
both both sides by 'string length' we get:

circumference
------------- = 2pi
string length

This tells us that there are exactly 2pi string lengths needed to go 
all the way around the circle.  Since each string length creates a 
central angle of 1 radian, there must also be 2pi radians in a circle.

Remember that when we used the string we figured that there were a 
little more than 6 radians in a circle.  What's the numerical value of 
2pi?  Using 3.14, we get 2 * 3.14 = 6.28 radians in a circle.  That's
a little more than 6, as we predicted!

Does that help clear things up?  If not, you might also find some
useful information in the Dr. Math archives:

  Degrees and Radians Explained
    http://mathforum.org/library/drmath/view/55450.html 

  From Degrees to Radians
    http://mathforum.org/library/drmath/view/57783.html 

  Are Angles Dimensionless?
    http://mathforum.org/library/drmath/view/64034.html 

Enjoy,

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/