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### 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

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

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

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/
```
Associated Topics:
High School Conic Sections/Circles

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