Angular to arc distance calculator

Enter the wheel radius and its signed rotation angle in radians, and the calculator applies the direct relationship between angular displacement and arc length. The result uses the same length unit as the radius, so a radius entered in metres produces metres, while millimetres produce millimetres. It is useful for wheels, drums, pulleys, rollers, encoders, and other rotating parts when slip is ignored and the rotation axis remains fixed.

Convert wheel rotation into travelled distance

A rolling wheel turns angular motion into linear motion at its rim. To calculate that travel, provide the wheel radius and the total rotation angle measured in radians. The calculator multiplies them using arc distance = radius × rotation angle. Because a radian is dimensionless, the answer keeps the radius's length unit. For example, entering a radius in metres returns metres; entering centimetres returns centimetres. Use the effective rolling radius, measured from the axle centre to the surface that contacts the ground or driven material. The angle should be the accumulated rotation over the interval you care about, not merely the wheel's final orientation modulo one revolution. One complete revolution is 2π radians, two revolutions are 4π radians, and so on. The calculation assumes rolling without slip. If a tyre skids, a belt slides, or the wheel deforms significantly under load, measured ground travel can differ from the ideal arc distance returned here.

Understand signs, radians, and units

The rotation input is signed. A positive angle produces a positive arc distance, while a negative angle produces a negative distance that can represent reverse rotation or travel in the opposite chosen direction. Radius, however, describes a physical magnitude and cannot be negative; the calculator reports an input error if a negative radius is supplied. A zero radius is valid mathematically and returns zero distance. Make sure the angle is already in radians. If your measurement is in degrees, convert it by multiplying degrees by π/180 before using this capability. If an encoder reports revolutions, multiply the revolution count by 2π. Keep unit handling consistent: do not enter a radius in millimetres and interpret the result as metres without conversion. The returned value is an ideal signed displacement at the circumference. If your application needs an unsigned path length regardless of direction changes, sum the absolute angular movement for each segment before multiplying, rather than using only the net signed rotation accumulated across the entire motion.

Apply the result to wheels, encoders, and rollers

This calculation is a useful bridge between rotary sensors and linear systems. For a wheel encoder, first convert pulse counts into radians using the encoder's counts per revolution, then pass the accumulated angle and calibrated rolling radius to obtain estimated travel. For a conveyor roller or winch drum, use the radius at the active contact layer. Be careful with drums that build up cable, paper, or material, because their effective radius changes as layers accumulate; a single fixed-radius calculation is accurate only for an interval where that change is negligible. Tyres also have an effective rolling radius that may differ from the unloaded geometric radius due to pressure, load, and deformation. Calibration against a known distance can improve practical estimates. The formula does not account for slip, backlash, wheel spin, steering geometry, or surface compression, so those effects should be modelled separately when accuracy demands it. For automated use, the API performs the same deterministic multiplication and costs $0.002 per request, making repeated calculations stable and easy to audit.