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Degrees, Radians, and Arc Length Solutions

• Circles have angles and arcs that can be measured.

• An arc is a portion of the circumference of a circle.

• The central angle of a circle is an angle whose vertex is the center of the circle.
• 
The angles of a circle can be measured in degrees or in radians, $\theta .$

• A radian is a unit of angle measure for a circle.

Example 1

Convert $150°$ to radians using π.

$\begin{array}{rcl}rad& =& deg·\frac{\mathrm{\pi }}{180}\\ rad& =& 150·\frac{\mathrm{\pi }}{180}=\frac{15\cancel{0}\mathrm{\pi }}{18\cancel{0}}\\ rad& =& \frac{5\mathrm{\pi }}{6}\end{array}$

Example 2

Convert the radian measure,$\frac{\mathrm{\pi }}{2}$, to a degree measure.

$$\begin{gathered} deg=rad·\frac{180}{\mathrm{\pi }} \\ deg=\frac{\mathrm{\pi }}{2}·\frac{180}{\mathrm{\pi }}=\frac{180\cancel{\mathrm{\pi }}}{2\cancel{\mathrm{\pi }}} \\ deg=90° \end{gathered}$$

Example 3

Determine the arc length of a circle with a $225°$ central angle and a radius of 7.

$$\begin{gathered} r=7 \\ rad =225·\frac{\mathrm{\pi }}{180}=\frac{225\mathrm{\pi }}{180}=\frac{5\mathrm{\pi }}{4} \\ \begin{array}{l}s=r \theta \\ s=7·\frac{5\mathrm{\pi }}{4}\\ s=\frac{35\mathrm{\pi }}{4}\end{array} \end{gathered}$$