Extension Lesson: Degrees, Radians, and Arc Length

Practice Solutions

For problems 1–10, if the problem is in degrees, convert it to radians using π.
If the problem is in radians, convert it to degrees.

$\frac{7 π}{6}$

$d e g = r a d \cdot \frac{180}{π} \frac{7 π}{6_{1}} \cdot \frac{180^{30}}{π} = 210 °$

$210 °$

$\frac{5 π}{12}$

$d e g = r a d \cdot \frac{180}{π} \frac{5 π}{12_{1}} \cdot \frac{180^{15}}{π} = 75 °$

$75 °$

$30 °$

$r a d = d e g \cdot \frac{π}{180} 30 \cdot \frac{π}{180_{6}} = \frac{π}{6}$

$\frac{π}{6}$

$\frac{3 π}{2}$

$d e g = r a d \cdot \frac{180}{π} \frac{3 π}{2_{1}} \cdot \frac{180^{90}}{π} = 270 °$

$270 °$

$360 °$

$r a d = d e g \cdot \frac{π}{180} 360^{2} \cdot \frac{π}{180_{1}} = 2 π$

$2 π$

$310 °$

$r a d = d e g \cdot \frac{π}{180} 310^{31} \cdot \frac{π}{180_{18}} = \frac{31 π}{18}$

$\frac{31 π}{18}$

$\frac{13 π}{20}$

$d e g = r a d \cdot \frac{180}{π} \frac{13 π}{20_{1}} \cdot \frac{180^{9}}{π} = 117 °$

$117 °$

$115 °$

$r a d = d e g \cdot \frac{π}{180} 115^{23} \cdot \frac{π}{180_{36}} = \frac{23 π}{36}$

$\frac{23 π}{36}$

$120 °$

$r a d = d e g \cdot \frac{π}{180} 120^{2} \cdot \frac{π}{180_{3}} = \frac{2 π}{3}$

$\frac{2 π}{3}$

$\frac{π}{4}$

$d e g = r a d \cdot \frac{180}{π} \frac{π}{4_{1}} \cdot \frac{180^{45}}{π} = 45 °$

$45 °$

Determine the arc length of a circle with the given central angle and radius.

$210 °, r = 3$

$r a d = d e g \cdot \frac{π}{180} r a d = 210^{7} \cdot \frac{π}{180_{6}} = \frac{7 π}{6} s = r θ s = 3 \cdot \frac{7 π}{6} = \frac{21 π}{6}$

$\frac{7 π}{2}$

$90 °, r = 8$

$r a d = d e g \cdot \frac{π}{180} r a d = 90 \cdot \frac{π}{180} = \frac{π}{2} s = r θ s = 8 \cdot \frac{π}{2}$

$4 π$

$300 °, r = 1$

$r a d = d e g \cdot \frac{π}{180} r a d = 300 \cdot \frac{π}{180} = \frac{5 π}{3} s = r θ s = 1 \cdot \frac{5 π}{3}$

$\frac{5 π}{3}$

$135 °, r = 10$

$r a d = d e g \cdot \frac{π}{180} r a d = 135 \cdot \frac{π}{180} = \frac{3 π}{4} s = r θ s = 10 \cdot \frac{3 π}{4} = \frac{30 π}{4}$

$\frac{15 π}{2}$

Find the missing value.

$d e g = 75 °, s = \frac{5 π}{6}, r =$

$r a d = d e g \cdot \frac{π}{180} r a d = 75 \cdot \frac{π}{180} = \frac{5 π}{12} s = r θ (\frac{12}{5 π}) \frac{5 π}{6} = r (\frac{5 π}{12}) (\frac{12}{5 π}) r = (\frac{12}{5 π}) \frac{5 π}{6}$

$r = 2$

$r = 5, s = \frac{25 π}{8}, d e g = °$

$\begin{array}{rcl} s & = & r θ \\ (\frac{1}{5}) \frac{25 π}{8} & = & 5 θ (\frac{1}{5}) \\ θ & = & \frac{5 π}{8} \\ \frac{5 π}{8} \cdot \frac{180}{π} & = & 5 \cdot \frac{45}{2} \end{array}$

$112.5 °$