Math101 · Independent Ontario learning library
Math101learn.math101.caArc 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.
Worked example
Representations and interpretation
Imagine unrolling the curved arc into a straight segment: its length depends linearly on both radius and angle in radians. A sector diagram shows the ratio $s/(2\pi r)=\theta/(2\pi)$.
Reasoning about variations
Two arcs with the same central angle but double radius have double length. An arc and its major counterpart sum to the full circumference, so the major length can be found by subtraction.
Common mistakes
How to check your work
- Verify $0\le s\le2\pi r$ for a minor/full-turn angle.
- Compare the arc-to-circumference ratio with the angle-to-full-turn ratio.
- Use the complementary major/minor arc sum.
Practice
- Find the length of a $60^\circ$ arc in a circle of radius $12$ cm.
- Use radians: find $s$ when $r=5$ m and $\theta=1.2$.
- A semicircle has radius $7$. Find its arc length.
Answers and brief solutions
Show answers
- $4\pi$ cm One sixth of circumference $24\pi$ is $4\pi$.
- $6$ m $s=r\theta=5(1.2)$.
- $7\pi$ A semicircle is half of circumference $14\pi$.
Synthesis and transfer
The distance travelled by a point on a rotating wheel is an arc length; using radians makes the product $r\theta$ directly consistent with linear units.
A wheel of radius $0.35$ m turning through $5$ radians moves a point on its rim through $s=r\theta=1.75$ m, provided there is no slipping. The same calculation in degrees would first convert the turn because the direct product formula assumes radians. Dividing by the full circumference shows that the angle occupies $5/(2\pi)$ of a revolution, offering a second route to the same distance. Arc length changes linearly with radius and angle, unlike sector area, which depends on radius squared. Dimensional analysis leaves metres after the dimensionless radian measure multiplies the radius.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Find the length of a $60^\circ$ arc in a circle of radius $12$ cm.
- One sixth of circumference $24\pi$ is $4\pi$.
End of lesson
Nice work making it this far.
Understanding grows through return visits. Save this lesson, try the practice, or continue when you are ready.
