Polar arc length is one of 2 polar formulas in the calculus section of this library, and it is used at ap · university level.
Why polar arc length works
A small change in the angle carries the point sideways through the radius times that angle change, while a change in the radius pushes it straight outward. Those two motions are perpendicular, so the true step is the hypotenuse, and pulling the angle step out of it produces the integrand.
What each symbol means
$r$ is radial distance and $\theta$ the polar angle.
Polar arc length: when it holds
$r$ and $dr/d\theta$ should be continuous on the interval and angles use radians.
When it stops applying
If the radius or its rate of change jumps or blows up, the integral has to be split or treated as improper. A spiral whose radius is one over the angle runs out to infinite distance as the angle approaches 0, so its length from there is not finite.
Polar arc length: a worked example
For $r=2$, $0\le\theta\le\pi$, $L=\int_0^\pi2d\theta=2\pi$.
The mistake to avoid
What people do: Integrating the radius against the angle and ignoring how the radius changes.
Why it goes wrong: That measures only the sideways sweep, so any curve whose radius grows or shrinks is reported as shorter than it really is.
Do this instead: Differentiate the radius with respect to the angle, square it, and add it under the root alongside the squared radius.
Polar 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 radial distance and $\theta$ the polar angle.
- Check the conditions before substituting. $r$ and $dr/d\theta$ should be continuous on the interval and angles use radians.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Polar
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where polar arc length comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Integration Techniques — the lesson behind this formula: choose substitution, parts, or simplification from structure.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about polar arc length
What does the formula give for a circle of fixed radius?
The radius does not change, so the integrand is just that radius, and a full turn returns 2π times it, which is the familiar way around.
Why does this resemble the parametric formula?
It is the parametric formula in disguise. Writing the coordinates in terms of the radius and the angle and simplifying produces exactly this integrand.
Should I worry about a negative radius?
The radius is squared inside the root, so length is unaffected. The only concern is whether the angle interval draws the curve once.
What if the curve closes before a full turn?
Stop at the angle where it closes up. Carrying on adds the very same length a second time.