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Arc Length

Portion of a circle's circumference.

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Arc Length

What if you were given the angle measure of a circle's arc? How could you find the length of that arc? After completing this Concept, you'll be able to find an arc's length, or its portion of a circle's circumference.

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Arc Length CK-12

Guidance

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}=\frac{m \widehat{AB}}{360^\circ} \cdot \pi d or \frac{m \widehat{AB}}{360^\circ} \cdot 2 \pi r .

Example A

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

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 \frac{1}{6} of that \left(\frac{60^\circ}{360^\circ}=\frac{1}{6}\right) . Therefore, the length of \widehat{PQ} is \frac{1}{6} of the circumference. length of \widehat{PQ}=\frac{1}{6} \cdot 2 \pi (9)=3 \pi units.

Example B

The arc length of a circle is \widehat{AB}=6 \pi and is \frac{1}{4} the circumference. Find the radius of the circle.

If 6 \pi is \frac{1}{4} 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 .

24 \pi &= 2 \pi r\\12 \ units &= r

Example C

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

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

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

Arc Length CK-12

Guided Practice

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

1.

2.

3. 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.

Answers:

1. Use the Arc Length formula.

\widehat{PQ}&=\frac{135}{360}\cdot 2 \pi (12)\\ \widehat{PQ}&=\frac{3}{8}\cdot 24 \pi \\ \widehat{PQ}&=9\pi

2. Use the Arc Length formula.

\widehat{PQ}&=\frac{360-260}{360}\cdot 2 \pi (144)\\ \widehat{PQ}&=\frac{5}{18}\cdot 288 \pi \\ \widehat{PQ}&=80\pi

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

Practice

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.

Vocabulary

chord

chord

A line segment whose endpoints are on a circle.
circle

circle

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

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

pi

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

radius

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

Arc

An arc is a section of the circumference of a circle.
arc length

arc length

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

Circumference

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

Dilation

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

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

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.

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