Arc Length of a Sector
Simple Explanation
The length of an arc — a curved portion of a circle's circumference — is found by s = rθ, where θ is the central angle in radians and r is the radius.
Why Do We Need It?
This directly connects an angle to a real physical length along a curve, essential for anything involving circular paths, from measuring pipes to designing curved roads.
See It
A circle sector with a 90-degree central angle, showing the two radii and the curved arc s between them
Formula
Arc Length
s = rθ (θ in radians)
The length of an arc equals the radius times the angle it subtends, when that angle is measured in radians.
- s
- — the arc length
- r
- — the radius of the circle
- θ
- — the central angle subtended by the arc, in radians
When to use it: Whenever you need the length of a curved arc from its radius and central angle.
Worked Example
Find an arc length
A circle has radius 8 cm. Find the arc length subtended by a central angle of 2 radians.
Why Does This Work?
By the definition of a radian, an angle of exactly 1 radian corresponds to an arc exactly r long — so an angle of θ radians, being θ times as large, corresponds to an arc exactly θ times as long, giving s = rθ directly.
Real-Life Example
Measuring the path of a Ferris wheel car
An engineer wants to know how far a Ferris wheel car travels while the wheel rotates through a given angle.
Using s = rθ with the wheel's radius and the rotation angle (in radians) gives the exact arc-length distance traveled by the car.
Practice
A circle has radius 6. Find the arc length for a central angle of 1.5 radians.
MediumCommon mistake
Using θ in degrees directly in s = rθ without converting to radians first — this formula only works correctly when θ is measured in radians.
Quick Review
- s = rθ, with θ in radians.
- Always convert θ to radians first if it is given in degrees.
- Comes directly from the definition of a radian.