Arc Formulas

Date: 05/08/2003 at 12:03:04
From: Rick
Subject: Remembering formula to use

I am trying to determine the angle of an arc from the radius and arc 
length. The radius is 630 and the arc length is 66.82.

How can I remember the formula?

Date: 05/08/2003 at 14:37:31
From: Doctor Ian
Subject: Re: Remembering formula to use

Hi Rick,

Here's how I remember it. If you have a circle with radius 1, the
circumference of the circle will be 2*pi, which is also the number of
radians in the circle.  

If I double the radius of the circle, I double the circumference, but
the number of radians stays the same. If I triple the radius, I triple 
the circumference, and so on. In general, if the radius of the circle 
is R, then the circumference is 2*pi*R, but the number of radians 
stays the same.  

So

  (radius of circle) * (angle in radians) = (arc length)

which is usually abbreviated

  R * theta = s

In your case, you know R and s, and you want to find theta, so you can
rearrange it to get

      theta = s/R

Note that the result will be in radians, not degrees. To convert, you
just have to remember that 2*pi radians (i.e., once around the circle)
is the same as 360 degrees (also once around the circle).  So to
convert to radians, you can multiply by this scale factor:

                360 degrees
  ___ radians * ----------- = ___ degrees
                 pi radians

If that's too complicated, here's another way to think about it. 
Suppose you have a circle with radius 660 cm.  The circumference of
the circle will be 

  circumference = 2 * 660 * pi

right? So that corresponds to 360 degrees. And you have some part of
that arc length, so you can set up and solve a proportion:

    ? degrees        66.82 cm 
   ----------- = -----------------
   360 degrees   (2 * 660 * pi) cm

Consider some test cases to see why this works. If your arc is the
same as the circumference, you should end up with 360 degrees, and you
do. If your arc is half the circumference, you should end up with 180
degrees, and you do.  

Basically, the ratio of the arc length to the circumference is the
same as the ratio of the angle to the whole 360 degrees of the circle.

Does that make sense? 

If you're not sure how to solve the proportion, take a look at

   Flipping and Switching Fractions
   http://mathforum.org/library/drmath/view/58193.html 

Does this help? 

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