Math101 · Independent Ontario learning library
Math101Arc Length
Arc length is the distance along part of a circle’s circumference. It is the same fraction of $2\pi r$ as the central angle is of a full turn: $s=(\theta/360^\circ)2\pi r$ in degrees.
Arc length measures curved travel in wheels, tracks, gears, and circular design. It also gives radians their geometric meaning.
Intuition and core definition
Arc length is the distance along part of a circle’s circumference. It is the same fraction of $2\pi r$ as the central angle is of a full turn: $s=(\theta/360^\circ)2\pi r$ in degrees. When $\theta$ is in radians, the direct formula is $s=r\theta$.
Notation, language, and conditions
$s$ denotes arc length, $r>0$ radius, and $\theta$ the central angle intercepting the arc. The radian formula requires radians; inserting a degree number into $r\theta$ is a unit error. A minor arc has measure below $180^\circ$ and a major arc above it.
Why this idea matters
Arc length scales a circle's circumference by the fraction of a full turn represented by its central angle.
A dependable method
- Identify radius and central angle, including angle units.
- Convert degrees to radians or use the degree-fraction formula.
- Multiply the full circumference by the fraction of a turn.
- Keep $\pi$ for an exact answer and approximate only if asked.
- Attach linear units and compare with the full circumference.
