Arc length is one of 6 circles formulas in the geometry section of this library, and it is used at high school level.
Why arc length works
A radian is defined so that an angle of one radian cuts off a piece of rim exactly as long as the radius. An angle of several radians therefore cuts off that many radii worth of rim, so multiplying the radius by the angle is really just the definition of radian measure doing the work.
What each symbol means
$r$ is radius and $\theta$ is the central angle in radians.
Arc length: when it holds
$\theta$ must be in radians; for degrees use $s=(\theta/360^\circ)2\pi r$.
When it stops applying
The angle has to be measured at the centre of the circle. An inscribed angle sitting on the rim cuts off the same piece while measuring only half as much, so using it here would halve the answer.
Arc length: a worked example
For $r=6$ and $\theta=\pi/3$, $s=2\pi$.
The mistake to avoid
What people do: Feeding a degree measure straight into the product.
Why it goes wrong: With a radius of 6 and an angle of 60 degrees that gives 360, when the piece of rim is really about 6.28 units long.
Do this instead: Convert to radians first by multiplying the degrees by π and dividing by 180, or use the version that divides the angle by 360.
Arc length: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $r$ is radius and $\theta$ is the central angle in radians.
- Check the conditions before substituting. $\theta$ must be in radians; for degrees use $s=(\theta/360^\circ)2\pi r$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most geometry slips.
Where this formula fits
- Subject
- Geometry formulas — 30 entries in this library
- Topic
- Circles
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where arc length comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Equation of a Circle — the lesson behind this formula: read and build circle equations on the coordinate plane.
- Geometry Calculator — check your substitution and the value it produces.
- Study geometry — the subject guide that explains the ideas these formulas compress.
- Geometry Practice — questions that make you retrieve the formula instead of recognising it.
- All 30 geometry formulas — the full grouped reference, or the complete formula library.
Questions about arc length
How do I convert degrees into radians?
Multiply by π and divide by 180. That turns 60 degrees into π/3 and turns a straight angle of 180 degrees into π.
What does the formula give for a full turn?
An angle of 2π returns 2π times the radius, which is the whole way around the circle, and that is a useful consistency check.
Is this the same as the straight distance between the endpoints?
No. This measures the curved rim while a chord cuts straight across, so the curved measurement is always the larger of the two.
How do I recover the radius from a piece of rim and its angle?
Divide the length of the rim by the angle in radians. A rim length of 2π from an angle of π/3 gives a radius of 6.