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Geometry

Arc length

The length of part of a circle's edge, cut off by a central angle θ.

s=rθ(θ in radians), s=θ360×2πr(θ in degrees)

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

See it move

θ = 60°
  • θ = 60° = 1.0472 rad, which is 16.667% of a full turn
  • Arc length = 5.236; sector area = 13.09

What the symbols mean

s
the arc length: the distance along the curved edge
r
the radius of the circle
θ
the central angle at the centre of the circle, in degrees or radians (use the toggle)

When to use it

Use it for the curved edge of a slice: the distance along a bend in a road, the length of a pendulum's swing, the edge of a pizza slice, distances along the Earth's surface between places on the same meridian. Switch the calculator to radians if your angle is in radians; in radians the formula is simply s=rθ, which is the reason radians exist.

Worked example

A pendulum 10 cm long swings through an angle of 60°. How long is the arc its tip travels? (r = 10, θ = 60°)

  1. Start from the formula with θ in degrees.

    s=θ360·2⁢π⁢r

  2. Substitute r = 10, θ = 60° and work it out.

    s=60360·2⁢π·10
    ≈10.472

Answer s ≈ 10.47 cm

Why it works

An angle θ is the fraction θ/360 of a full turn, so the arc is that fraction of the whole circumference 2πr. A radian is defined as the angle whose arc equals the radius, which makes the fraction disappear.

Common mistakes

  • Putting degrees into s=rθ. With θ = 60, that gives 600 instead of about 10.47; convert to radians or use the degree form.
  • Using the chord (the straight line between the ends) instead of the arc.
  • Using the diameter instead of the radius.