Arc Length
Length of a circular arc (θ in radians).
Formula
Variables
Example
Radius 10, angle 1.5 rad: s = 10·1.5 = 15 units.
How it works
This formula gives the distance along the edge of a circle between two points, as long as the angle is measured in radians. It works because a radian is defined so that an angle of 1 radian subtends an arc whose length equals the radius, so the arc length scales directly with both radius and angle. Larger circles make longer arcs for the same angle, and larger angles sweep out more of the circumference.
When to use it
Use it whenever you know a circle’s radius and the central angle in radians and need the length of the curved part. It appears in geometry, trigonometry, physics, engineering, and any problem involving sectors, rotation, or circular motion.
Worked example, step by step
A circle has radius 7 cm and central angle 0.8 rad. Use s = rθ s = 7 × 0.8 s = 5.6 Final answer: the arc length is 5.6 cm.
Common mistakes to avoid
- Using degrees instead of radians
- Multiplying by diameter instead of radius
- Forgetting that the answer has the same length units as the radius
Did You Know?
Arc length is why radians are natural: in radians, arc length is simply radius times angle.
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