Degree Measure of a Central Angle

Date: 07/29/97 at 11:38:04
From: Tawny Woods
Subject: Trigonometry

Find the degree measure, to the nearest tenth, of a central angle 
whose intercepted arc measures 21 cm. in a circle of radius 4 cm.

Date: 08/30/97 at 12:22:42
From: Doctor Sonya
Subject: Re: Trigonometry

Dear Tawny,

To solve this problem, you first need to look at what information you 
have. All we know is the length of the intercepted arc and the radius 
of the circle. What else do we need to find to get the answer?

If we could somehow figure out what fraction of the circumference the 
intercepted arc is, wouldn't it make sense to say that the intercepted 
angle is the same fraction of 360 degrees, the total angular measure 
the circle?  

If you draw a picture of a circle with an intercepted arc, this idea 
becomes a little clearer.

To put the above idea into action, we need to figure out the 
circumference of the circle.  There is a very simple formula that says 
that if a circle has circumference C and diameter D, C = piD where 
"pi" is a constant equal to 3.14159... with a decimal that goes on 
forever.  Usually I just leave it as "pi" and then plug the value (to 
however many decimal places I need) at the end.  Your calculator might 
also have a "pi" button that would plug the value in for you 
automatically.

With this formula we can find the circumference of the circle. Say we 
calculate it to be c. Then the intercepted arc is 21/c of the circle.  
If c turns out to be, say, 30, then the arc is 21/30ths of the 
circumference.  

Now we want to find the central angle that is the same fraction of 
360.

To find this, call the measure of our central angle A.  A/360 will 
tell us what fraction A is of the total anglular measure. Since we 
want it it be equal to the first fraction we found, we can write:

     21/c = A/360

Once you know what c is, just plug it in, and the value for A 
shouldn't be too hard to find.  

What we just did is called setting up a ratio, and is one of the most 
useful techniques in math, especially in algebra and geometry!  

-Doctors Sonya and Kelli,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/