SOLUTION: A circle with circumference 12<font face="symbol">p</font> has an arc with a 189° central angle. What is the length of the arc?

We want the length of the red arc:

The easy way is with the proportion:



Let the arc length be s:



Reduce the fraction on the right



Cross-multiply:

40s = 252p

Divide both sides by 40

s = 

s = 

That's the exact answer.

The approximate answer is 19.792.

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There is a harder way which some teachers require students 
to use so they will learn the formula for the length of an arc,
and so they'll learn to convert degrees to radians, in cases
when just the angle in degrees and the radius are given.

The formula for the length of an arc is: 

s = q·r where font face="symbol">q is
measured in radians.

So we need to do two things:

1. Convert q = 189° to radians.
2. Find the radius of the circle.

To convert 189° to radians we multiply it by  

 q =  = 

To find the radius of the circle, we use

 C = 2pr

12p = 2pr

Divide both sides by 2p

 = 

6 = r

s = q·r

s = ·

s = ·

s = 

s = 

That's the exact answer.

The approximate answer is 19.792.

Edwin