The prior section,
Review of Circles (and related concepts),
reviews definitions:
circle, radius, diameter, circumference,
the irrational number pi,
central angles.
Let $\,c\,$ denote the circumference of a circle. Note: circumference is arc length with a central angle of $\,360^\circ\,$. Computing arc length is simple: find the desired fraction ($\frac{\text{part}}{\text{whole}}$) of the circumference! Central angles are used to determine the desired fraction. Be sure to use the same angle units for the ‘part’ and the ‘whole’, so that units cancel (as needed). Here are examples: $ \begin{align} &\text{the length of an arc with central angle $\,180^\circ\,$}\cr &\qquad= \frac{180^\circ}{360^\circ}\cdot c\cr &\qquad= \frac{180}{360}\cdot c \qquad\qquad \text{(units cancel)}\cr &\qquad= \frac{1}{2}c \end{align} $ $ \begin{align} &\text{the length of an arc with central angle $\,90^\circ\,$}\cr &\qquad= \frac{90^\circ}{360^\circ}\cdot c\cr &\qquad= \frac{1}{4}c \end{align} $ $ \begin{align} &\text{the length of an arc with central angle $\,60^\circ\,$}\cr &\qquad= \frac{60^\circ}{360^\circ}\cdot c\cr &\qquad= \frac{1}{6}c \end{align} $ 
Arc length is a desired fraction of the circumference: $$ \begin{align} &\text{arc length}\cr\cr &\qquad = (\text{desired fraction})(\text{circumference})\cr\cr &\qquad = \left(\frac{\small\text{central angle corresponding to desired arc length}}{\small\text{central angle corresponding to circumference}}\right)(\text{circumference})\cr\cr \end{align} $$ 
Define $\,r\,$, $\,\theta\,$, and $\,s\,$ as shown at right:
providing you use the correct formula. Keep reading! The units for $\,r\,$ and $\,s\,$ are (the same) length unit: e.g., feet, meters, miles, centimeters. For example, if $\,r\,$ is given in (say) feet, then a computed value for $\,s\,$ will also be feet. Or, if $\,s\,$ is given in (say) centimeters, then a computed value for $\,r\,$ will also be centimeters. 

In degree measure, one complete revolution is $\,360^\circ\,$. The circumference of a circle with radius $\,r\,$ is $\,2\pi r\,$. Circumference takes the unit of $\,r\,$, since the real number $\,2\pi\,$ has no units. Let $\,\theta\,$ be a real number, without units, that represents the degree measure of a central angle. For example, if $\,\theta = 60\,$, then $\,\theta^\circ = 60^\circ\,$. The arc length corresponding to $\,\theta\,$ is computed as follows: $$ \begin{align} s& = \left(\frac{\small\text{central angle corresponding to desired arc length}}{\small\text{central angle corresponding to circumference}}\right)(\text{circumference})\cr\cr &= \frac{\theta^\circ}{360^\circ}(2\pi r)\cr \end{align} $$ It may be easiest to remember the formula in one of these forms, $$\,s = \frac{\theta^\circ}{360^\circ}(2\pi r)\qquad \text{or} \qquad s = \frac{\theta}{360}(2\pi r),\,$$ since they emphasize that arc length is just a desired fraction of the circumference. However, the formula can be simplified: $$ \begin{align} s &= \frac{\theta^\circ}{360^\circ}(2\pi r)\cr\cr & = \frac{\theta}{360}(2\pi r) \qquad\qquad \text{(the degree units cancel)}\cr\cr & = \frac{\theta}{180}\cdot \pi r \end{align} $$ Thus, a simplified formula for arc length is: $$ s = \frac{\pi r\theta}{180} $$ Equivalently, we can use a proportion: $$ \begin{gather} \frac{\text{part angle}}{\text{whole angle}} = \frac{\text{part arc length}}{\text{whole arc length}}\cr\cr \frac{\theta^\circ}{360^\circ} = \frac{s}{2\pi r}\cr\cr \end{gather} $$ Cancel the degree units on the left, and then crossmultiply and solve for $\,s\,$: $$ \begin{gather} 2\pi r\theta = 360s\cr\cr s = \frac{2\pi r\theta}{360}\cr\cr s = \frac{\pi r\theta}{180} \end{gather} $$ 
ARC LENGTH FORMULA WHEN CENTRAL ANGLE IS MEASURED IN DEGEES There is a subtle but annoying issue that arises in the arc length formula using degree measure, as follows: Say you want the arc length corresponding to a central angle of $\,60^\circ\,$. You want to use the variable $\,\theta\,$ to ‘hold’ this central angle information. You have two choices:
At left, choice (2) was used, yielding the formula: $$ s = \frac{\pi r\theta}{180} $$ For choice (1), the degree units can't be cancelled, since the unit is ‘trapped’ inside the variable $\,\theta\,$. In this case, the formula becomes: $$ s = \frac{\pi r\theta}{180^\circ} $$ Of course, when you put
and use the appropriate formula! 
In radian measure, one complete revolution is $\,2\pi\,$. Recall that the radian measure of an angle is a real number. Radian measure has no units. Consequently, there are no annoying issues in the formula for arc length when the central angle is measured in radians. Hooray! The circumference of a circle with radius $\,r\,$ is $\,2\pi r\,$. Circumference takes the unit of $\,r\,$, since the real number $\,2\pi\,$ has no units. Let $\,\theta\,$ be the radian measure of a central angle. The arc length corresponding to $\,\theta\,$ is computed as follows: $$ \begin{align} s& = \left(\frac{\small\text{central angle corresponding to desired arc length}}{\small\text{central angle corresponding to circumference}}\right)(\text{circumference})\cr\cr &= \frac{\theta}{2\pi}(2\pi r)\cr\cr & = \theta r \qquad \text{(cancel $\,2\pi\,$)} \end{align} $$ Thus, the formula for arc length (written in the most conventional way) is: $$ s = r\theta $$ Equivalently, we can use a proportion: $$ \begin{gather} \frac{\text{part angle}}{\text{whole angle}} = \frac{\text{part arc length}}{\text{whole arc length}}\cr\cr \frac{\theta}{2\pi} = \frac{s}{2\pi r}\cr\cr \end{gather} $$ Crossmultiply and solve for $\,s\,$: $$ \begin{gather} 2\pi r\theta = 2\pi s\cr\cr s = r\theta \end{gather} $$ 
ARC LENGTH FORMULA WHEN CENTRAL ANGLE IS MEASURED IN RADIANS
$$
s = r\theta
$$
and simple formula! As an alternative to the degreemeasure derivation given above, we can ‘convert’ the formula $\,s = r\theta\,$ to one that works for central angles measured in degrees, as follows: First, convert $\,\theta^\circ\,$ to radian measure. Since $\pi\,$ radians equals $\,180^\circ\,$, we have: $$ \,1 = \frac{\pi}{180^\circ}$$ Therefore: $$\theta^\circ \ =\ \theta^\circ\cdot 1 \ =\ \theta^\circ\cdot\frac{\pi}{180^\circ} \ =\ \frac{\pi\theta}{180}$$ Thus, $\,\displaystyle\frac{\pi\theta}{180}\,$ is the radian measure of $\,\theta^\circ\,$. Subsitution of this radian measure into the formula $\,s = r\theta\,$ gives $$ s \ =\ r\left(\frac{\pi\theta}{180}\right) \ =\ \frac{\pi r\theta}{180} $$ which, of course, is the same formula derived above. Thus, we see that the ‘$\frac{\pi}{180}\,$’ is just a conversion factor from degree to radian measure! 
The formula $\,s = r\theta\,$ gives a beautiful insight into why the
name ‘radian’ is used for radian measure.
The idea was discussed in an earlier section on
radian measure, but is
worth repeating here!
Rewrite $\,s = r\theta\,$ (where $\,\theta\,$ is the radian measure of an angle) as $\,\displaystyle\theta = \frac{s}{r}\,$. Now, think about the interpretation of the division problem $\displaystyle\,\frac{s}{r}\,$. The division problem $\displaystyle\,\frac{10}{2}\,$ answers the question ‘How many $2$'s are there in $\,10\,$?’ Answer: $5$ Similarly, the division problem $\displaystyle\,\frac{s}{r}\,$ answers the question: Answer: the radian measure of the angle! 

The radian measure of an angle gives the number of radiuses needed to subtend the angle! 
measure: the number of RADiuses IN an angle 
This knowledge makes it easy to visualize angles that are given in
radian measure. For example, suppose you want to visualize an angle with measure $\,2.5\,$ radians. Draw a circle of any size, as shown at right. Imagine cutting a string the length of the radius of the circle. Lay this string around the circle $\,2.5\,$ times: $\,1r + 1r + 0.5r = 2.5r\,$ Voila! You just laid off an angle with measure $\,2.5\,$ radians! 
What does a $\,2.5\,$ radian angle look like? Lay off $\,2.5\,$ radiuses! 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
