# Challenge problems: Arc length 2

Solve two challenging problems that ask you to find an arc measure using the arc length.

## Problem 1

In the figure below, the radius of circle $P$P is $10$10 units. The arc length of $\stackrel{\Huge{\frown}}{ABC}$ is $16\pi$16, pi.

**What is the arc measure of $\stackrel{\LARGE{\frown}}{AC}$, in degrees?**

We are trying to figure out the

*arc measure*of the arc that begins at point $A$A and ends at point $C$C.We need to know the total circumference in order to determine the arc measure for $\stackrel{\Huge{\frown}}{ABC}$.

$\begin{aligned}\text{Circumference} &= 2\pi r\\ &= 2\pi(10)\\ &= 20\pi \end{aligned}$

We know the length of $\stackrel{\Huge{\frown}}{ABC}$, so we set up a proportion to figure out its arc measure.

$\begin{aligned} \dfrac{\text{arc length}}{\text{circumference}} &= \dfrac{\text{arc measure}}{\text{degrees in a circle}}\\\\\\ \dfrac{16\pi}{20\pi} &= \dfrac{\text{arc measure}}{360^\circ}\\\\\\ \text{arc measure} &=360^\circ \cdot\dfrac{16\pi}{20\pi}\\\\\\ &=288^\circ \end{aligned}$

The arc measure of $\stackrel{\Huge{\frown}}{ABC}$ is $288^\circ$288, degree.

If we combine the major arc $\stackrel{\Huge{\frown}}{ABC}$ and the minor arc $\stackrel{\LARGE{\frown}}{AC}$, we have the entire circle.

$\begin{aligned}288^\circ+ \text{m}\stackrel{\LARGE{\frown}}{AC}\, = 360^\circ\\ \text{m}\stackrel{\LARGE{\frown}}{AC}\, = 72^\circ\\\end{aligned}$

**The measure of $\stackrel{\LARGE{\frown}}{AC}$ is $72^\circ$72, degree.**

## Problem 2

In the figure below, the radius of circle $P$P is $18$18 units. The arc length of $\stackrel{\LARGE{\frown}}{BA}$ is $14\pi$14, pi.

**What is the arc measure of $\stackrel{\LARGE{\frown}}{BC}$, in degrees?**

**Your answer should be**- an integer, like $6$6
- a
*simplified proper*fraction, like $3/5$3, slash, 5 - a
*simplified improper*fraction, like $7/4$7, slash, 4 - a mixed number, like $1\ 3/4$1, space, 3, slash, 4
- an
*exact*decimal, like $0.75$0, point, 75 - a multiple of pi, like $12\ \text{pi}$12, space, p, i or $2/3\ \text{pi}$2, slash, 3, space, p, i

We are trying to figure out the

*arc measure*of the arc that begins at point $B$B and ends at point $C$C.Let's calculate the full circumference first, then solve proportionally for the arc measure based on the arc length.

$\begin{aligned}\text{Circumference} &= 2\pi r\\ &= 2\pi(18)\\ &= 36\pi \end{aligned}$

We know the length of $\stackrel{\LARGE{\frown}}{BA}$, so we can set up a proportion to figure out its arc measure.

$\begin{aligned} \dfrac{\text{arc length}}{\text{circumference}} &= \dfrac{\text{arc measure}}{\text{degrees in a circle}}\\\\\\ \dfrac{14\pi}{36\pi} &= \dfrac{\text{arc measure}}{360^\circ}\\\\\\ \text{arc measure} &=\dfrac{14\pi}{36\pi}\cdot {360^\circ}\\\\\\ &=140^\circ \end{aligned}$

The measures of $\stackrel{\LARGE{\frown}}{BC}$ and $\stackrel{\LARGE{\frown}}{CA}$ add up to the measure of $\stackrel{\LARGE{\frown}}{BA}$.

$\text{m}\stackrel{\LARGE{\frown}}{BC}+76^\circ = 140^\circ$

**The arc measure of $\stackrel{\LARGE{\frown}}{BC}$ is $64^\circ$64, degree.**