HesapHanem

Arc Length (Radians) Calculator

Arc length from a radius and a radian angle.

Result

14.40

How it is calculated

Enter the circle's radius r in any length unit and the central angle θ in radians. The arc length is:

s = r × θ

The result is in the same unit as the radius, rounded to two decimals. The radius must be above zero and the angle cannot be negative. No upper limit is applied to the angle, so a value above 2π (about 6.283) simply means the arc wraps around the circle more than once; 7 radians on a radius of 12 gives 84.00. If your angle is in degrees, convert it first with radians = degrees × π / 180.

Worked example

A wheel of radius 12 cm turns through an angle of 1.2 radians. The arc length is 12 × 1.2 = 14.40 cm. As a check, a full turn (2π ≈ 6.283 radians) would cover 12 × 6.283 = 75.40 cm, the circumference, so 1.2 radians is a bit under a fifth of the way round.

Try this example →

Common mistakes

  • Entering the angle in degrees. Typing 90 instead of 1.5708 gives an arc about 57 times too long.
  • Entering the diameter instead of the radius, which doubles the answer.
  • Expecting the result in degrees or in angle units. It is a length in the radius's unit.
  • Using a negative angle to mean the opposite direction; the tool rejects it, so enter the size of the angle.
  • Rounding π to 3 when converting degrees to radians by hand, which makes the arc visibly too short on large radii.

Sources

About this calculator

When an angle is measured in radians, arc length reduces to a one-line formula: s = r × θ, because a radian is defined as the angle that subtends an arc equal in length to the radius — no conversion factor or π needed, unlike the degree version which requires s = r × θ × π/180.

This direct relationship is why engineers, machinists, and physics students prefer working in radians for anything involving rotation or circular motion — gear tooth spacing, pulley belt lengths, satellite orbital paths, and pendulum swing distances are all sized with this exact formula.

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Comments (1)

  • Editorial team

    One everyday use is belt or cable wrap on a pulley: the contact length is the radius times the wrap angle in radians. For a pulley of radius 5 cm wrapped through half a turn, enter 3.1416 and read about 15.71 cm. Since the result keeps the radius unit, keep the radius in the unit you want back. For a long wrap, remember the tool gives the curved part only, not any straight run between pulleys. What are you wrapping or rotating?

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