Key takeaway: Arc length is the distance measured along a curved piece of a circle, found by taking the fraction of the full turn that the piece covers times the circumference.
Grade 10 – GeometryG-C.B.5Geometry and Measurement
Definition of Arc Length
Arc length answers how far you would walk along a curved edge of a circle. A circle’s whole edge has length equal to the circumference, \(2\pi r\). A piece that covers part of the turn has the same part of that length.
If the piece is cut off by a central angle of \(\theta\) degrees, it covers \(\frac{\theta}{360}\) of the circle. So the distance is that fraction multiplied by \(2\pi r\). Unlike the degree measure of the piece, this value carries units such as feet or centimeters, and it grows with the radius.
Formula and notation
The arc length \(s\), for radius \(r\) and central angle \(\theta\) in degrees, is:
\[ s = \frac{\theta}{360}\cdot 2\pi r \]
If \(\theta\) is measured in radians, the formula simplifies to \(s = r\theta\). Leave the answer in terms of \(\pi\) for an exact value, or round for a decimal.
Key facts
- Doubling the central angle doubles the length, for the same radius.
- Doubling the radius doubles the length, for the same angle.
- A full circle at 360 degrees gives the whole circumference.
- The units of the answer match the units of the radius.
Arc Length example
A Ferris wheel has a radius of 35 ft. A rider moves through a central angle of \(72^{\circ}\). How far along the rim does the rider travel?
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Fraction of the turn: \(\frac{72}{360} = \frac{1}{5}\).
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Circumference: \(2\pi(35) = 70\pi\) ft.
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Distance: \(\frac{1}{5}\cdot 70\pi = 14\pi \approx 44.0\) ft.
The rider travels \(14\pi \approx 44.0\) ft along the rim.
Common mistake
Watch out
Kofi finds the arc length with the diameter by mistake and computes \(\frac{72}{360}\cdot 2\pi(70) = 28\pi\) ft.
Why it is wrong
The formula uses the radius, which is 35 ft, not the 70-ft diameter. Using 70 doubles the answer. Always read the problem for “radius” or “diameter” before substituting into \(2\pi r\).
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement