Why 6.28 Radians?

Date: 10/07/98 at 21:16:44
From: Jason 
Subject: Radians

Why is a circle divided into approximately 6.28 radians?

Date: 10/08/98 at 13:18:41
From: Doctor Peterson
Subject: Re: Radians

Hi, Jason,

A complete circle, 360 degrees, is exactly 2 pi radians. Do you 
recognize that from the formula for the circumference of a circle, 
2 pi r? That's because the number of radians in an angle is defined as 
the length of  the arc of a unit circle subtended by the angle:

                       ***********
                 ******           ****/*
              ***                    /  ***
           ***                      /      ***
          *                        /          *  L
        **                        /            **
       *                         /               *
      *                         /                 *
     *                         /                   *
    *                         /                     *
    *                        / A                    *
    *                       *------------------------
    *                                   1           *
    *                                               *
     *                                             *
      *                                           *
       *                                         *
        **                                     **
          *                                   *
           ***                             ***
              ***                       ***
                 ******           ******
                       ***********

The radian measure of the angle A is defined as the length of the arc 
L. You can think of radians as just using a flexible tape measure to 
measure around the circle - the most natural way to measure an angle.

So if A is the whole circle, its measure is the whole circumference, 
or 2 times pi, which is about 6.28.

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