Cambridge O Level Additional Mathematics · Syllabus 4037 · Circular Measure
Arc Length
What is Arc Length?
The distance measured along the curved edge of a circle between two points on it, equal to the radius multiplied by the central angle the arc subtends when that angle is measured in radians, written s = r theta. It is a length and therefore carries linear units such as centimetres. The formula is a direct rearrangement of the definition of the radian and is not supplied in the examination formula list, so it must be recalled; it is invalid for an angle given in degrees.
This definition is part of the Circular Measure chapter in Cambridge O Level Additional Mathematics.
Arc Length in context
A radian is the angle for which the arc it cuts off equals the radius, defined as the ratio \(\theta=\dfrac{s}{r}\) of arc length to radius. Because it is one length divided by another, it carries no unit and is a pure number, so a full turn equals \(2\pi\) radians and \(180^\circ=\pi\) radians is the single conversion anchor. Two formulas follow directly from that ratio with no extra constants: arc length \(s=r\theta\) and sector area \(A=\tfrac12r^2\theta\). Both are valid only when \(\theta\) is in radians; neither formula is printed in the examination formula list, so both must be recalled.
Questions students ask about Arc Length
What is a radian?
A radian is the angle for which the arc it cuts off equals the radius, defined as \(\theta=\dfrac{s}{r}\), the ratio of arc length to radius. Because it is one length divided by another it carries no unit and is a pure number, roughly \(57.3^\circ\). A full turn is \(2\pi\) radians, and the anchor identity is \(180^\circ=\pi\) radians; every other conversion is a multiple or fraction of that.
Why does using degrees in \(s=r\theta\) or \(A=\tfrac12r^2\theta\) give the wrong answer?
Both formulas come directly from the definition \(\theta=s/r\), which measures the angle in radii, so they are only valid when \(\theta\) is in radians. A degree value is about \(57.3\) times too large, so an arc length or sector area calculated with degrees is roughly \(57.3\) times too big, while still looking like a plausible number. Convert the angle to radians first, before substituting into either formula, and check a sector area against \(\pi r^2\) — it should never exceed it.