Cambridge O Level Additional Mathematics · Syllabus 4037 · Circular Measure
Sector Area
What is Sector Area?
The area of the pie-shaped region of a circle enclosed by two radii and the arc between them, equal to one half of the radius squared multiplied by the central angle in radians, written A = one half r squared theta. Because it is an area it carries square units such as square centimetres. Like the arc-length formula it follows from taking the fraction theta over 2 pi of the whole disc, it is not supplied in the examination formula list, and it is invalid unless the angle is in radians.
This definition is part of the Circular Measure chapter in Cambridge O Level Additional Mathematics.
Sector Area 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.
Common mistakes with Sector Area
- Using \(\tfrac12r^2\theta\) with \(\theta\) in degrees. Why it fails Same cause, same factor. The sector area comes out about \(57.3\) times too large, and will usually exceed \(\pi r^2\) — a region bigger than the circle containing it. Fix Convert first; then compare the answer with \(\pi r^2\).
Questions students ask about Sector Area
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.