# 6.11: Arc Length

- Page ID
- 5026

Portion of a circle's circumference.

One way to measure arcs is in degrees. This is called the “**arc** measure” or “degree measure” (see Arcs in Circles). Arcs can also be measured in length, as a portion of the **circumference**. **Arc length** is the length of an arc or a portion of a **circle**’s circumference. The arc length is directly related to the degree arc measure.

**Arc Length Formula:** The length of \(\widehat{AB}=\dfrac{m\widehat{AB}}{360^{\circ}}\cdot \pi d\) or \(\dfrac{m\widehat{AB}}{360^{\circ}}\cdot 2\pi r\).

What if you were given the angle measure of a circle's arc? How could you find the length of that arc?

Example \(\PageIndex{1}\)

Find the arc length of \(\widehat{PQ}\) in \(\bigodot A\). Leave your answers in terms of \(\pi\).

**Solution**

Use the Arc Length formula.

\(\begin{aligned} \widehat{PQ}&=\dfrac{135}{360}\cdot 2\pi (12) \\ \widehat{PQ}&=\dfrac{3}{8}\cdot 24\pi \\ \widehat{PQ}&=9\pi \end{aligned}\)

Example \(\PageIndex{2}\)

A typical large pizza has a **diameter** of 14 inches and is cut into 8 pieces. Think of the crust as the circumference of the pizza. Find the * length* of the crust for the entire pizza. Then, find the length of the crust for one piece of pizza if the entire pizza is cut into 8 pieces.

**Solution**

The entire length of the crust, or the circumference of the pizza, is \(14\pi \approx 44 in.\) In \(\dfrac{1}{8}\) of the pizza, one piece would have \(\dfrac{44}{8}\approx 5.5 inches\) of crust.

Example \(\PageIndex{3}\)

Find the length of \(\widehat{PQ}\). Leave your answer in terms of \(\pi\).

**Solution**

In the picture, the central angle that corresponds with \(\widehat{PQ}\) is \(60^{\circ}\). This means that \(m\widehat{PQ}=60^{\circ}\). Think of the arc length as a portion of the circumference. There are \(360^{\circ}\) in a circle, so 60^{\circ}\) would be 16 of that (\(60^{\circ}360^{\circ}=16\)). Therefore, the length of \(\widehat{PQ}\) is 16 of the circumference. \text{length of } \widehat{PQ}=16\cdot 2\pi (9)=3\pi \text{ units}\).

Example \(\PageIndex{4}\)

The arc length of a circle is \(\widehat{AB}=6\pi\) and is 14 the circumference. Find the **radius** of the circle.

**Solution**

If 6\pi is 14 the circumference, then the total circumference is \(4(6\pi )=24\pi\). To find the radius, plug this into the circumference formula and solve for r.

\(\begin{aligned} 24\pi =2\pi r \\ 12 \text{ units }=r \end{aligned}\)

Example \(\PageIndex{5}\)

Find the measure of the central angle or \(\widehat{PQ}\).

**Solution**

Let’s plug in what we know to the Arc Length Formula.

\(\begin{aligned}15\pi &=m\widehat{PQ}360^{\circ}\cdot 2\pi (18) \\ 15&=m\widehat{PQ}10^{\circ} \\ 150^{\circ}&=m\widehat{PQ}\end{aligned}\)

## Review

Find the arc length of \(\widehat{PQ}\) in \(\bigodot A\). Leave your answers in terms of \(\pi\).

Find \(PA\) (the radius) in \(\bigodot A\). Leave your answer in terms of \(\pi \).

Find the central angle or \(m\widehat{PQ}\) in \(\bigodot A\). Round any decimal answers to the nearest tenth.

## Review (Answers)

To see the Review answers, open this PDF file and look for section 10.9.

## Vocabulary

Term | Definition |
---|---|

chord |
A line segment whose endpoints are on a circle. |

circle |
The set of all points that are the same distance away from a specific point, called the .center |

diameter |
A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius. |

pi |
(or \(\pi\) ) The ratio of the circumference of a circle to its diameter. |

radius |
The distance from the center to the outer rim of a circle. |

Arc |
An arc is a section of the circumference of a circle. |

arc length |
In calculus, arc length is the length of a plane function curve over an interval. |

Circumference |
The circumference of a circle is the measure of the distance around the outside edge of a circle. |

Dilation |
To reduce or enlarge a figure according to a scale factor is a dilation. |

radian |
A radian is a unit of angle that is equal to the angle created at the center of a circle whose arc is equal in length to the radius. |

Sector |
A sector of a circle is a portion of a circle contained between two radii of the circle. Sectors can be measured in degrees. |

## Additional Resources

Interactive Element

Video: Arc Length Principles - Basic

Activities: Arc Length Discussion Questions

Study Aids: Circumference and Arc Length Study Guide

Practice: Arc Length

Real World: How Far Is It to London?