Arc Length

Arc length is radius times angle — and this little product is no coincidence, it is the very definition of the radian: one radian is the angle whose arc exactly equals the radius. That makes s = rθ the reason radians exist at all. The name was coined around 1870 by James Thomson (brother of Lord Kelvin), but the idea of measuring angles by arc goes back to Roger Cotes in 1714. The relation holds for angles from 0 up to 2π, where the arc becomes the full circumference 2πr. Enter degrees and formula.expert converts them to radians before multiplying.

Engineers lean on this constantly. A highway curve of radius 200 m that turns the road through 30° (0.524 rad) contains 200 × 0.524 ≈ 105 m of pavement; a gear tooth's working surface, a satellite's ground track, and the belt wrap on a pulley are all measured the same way.