Arc Length
In short: Arc length is the real distance you would walk along a bent stretch of a circle’s rim, equal to the share of the circumference matching the opening at the center.
GeometryGeometryG-C.B.5
What is Arc Length?
Walk along a circular track from one marker to another and count the steps: that distance is the arc length. It is a slice of the circumference, and the slice size comes straight from the central angle \(\theta\) in degrees.
Since the whole circle is \(360^{\circ}\) and its circumference is \(2\pi r\), the formula is \(s = \frac{\theta}{360} \times 2\pi r\). With the angle in radians the formula simplifies to \(s = r\theta\). The result is a length in feet, meters or other units, unlike the degrees used to measure the piece.
How to spot it
- A problem gives a radius and a central angle and asks how far to travel along a curve.
- A curved fence, track, pizza crust, clock hand tip or wheel rim is described with an opening angle.
- The answer must be in units such as feet or centimeters, and an exact value with \(\pi\) is often accepted.
- A fraction of a circle is mentioned and the circumference must be shared in that proportion.
The method
- Write the fraction of the circle: the central angle over \(360\).
- Compute the full circumference \(2\pi r\) from the radius.
- Multiply the fraction by the circumference, and keep \(\pi\) exact if the problem allows it.
- Convert to a decimal at the end and attach the unit.
Arc Length: solved example
Problem
A curved garden border follows a circle of radius 15 feet and spans a central angle of \(72^{\circ}\). How much edging is needed?
Solution
The fraction of the circle is \(\frac{72}{360} = \frac{1}{5}\). The circumference is \(2\pi(15) = 30\pi\) feet.
Arc length: \(\frac{1}{5} \times 30\pi = 6\pi \approx 18.85\). The border needs \(6\pi \approx 18.85\) feet of edging.
Try it
Question 1
A bicycle wheel with a 12-inch radius rolls until it has turned \(150^{\circ}\). How far has a speck on the tire traveled along the rim? Give an exact value and a decimal.
Show the answer
\(\frac{150}{360} \times 24\pi = 10\pi \approx 31.4\) inches.
Question 2
A bent railing follows a circle of radius 10 cm and measures \(5\pi\) cm. What opening does it span at the center?
Show the answer
\(5\pi = \frac{\theta}{360} \times 20\pi\), so \(\theta = 90^{\circ}\).
Arc Length vs. Arc
The measure of a piece of a circle is an angle in degrees and depends only on the opening at the center. The length is a distance in feet or centimeters and also depends on the radius. A \(72^{\circ}\) piece is always \(72^{\circ}\), but its length is 18.85 feet on a 15-foot circle and 37.7 feet on a 30-foot one.
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