Cambridge O Level Additional Mathematics · Syllabus 4037 · Circular Measure
Circular Segment
What is Circular Segment?
The region of a circle cut off by a chord, bounded by that chord and one of the two arcs it separates. Its area is found by removing from the sector the triangle formed by the two radii and the chord, giving one half r squared multiplied by the quantity theta minus sine theta, where theta is the central angle in radians and the sine is evaluated on that same radian value. No radius lies on a segment's boundary, so the perimeter of a segment is the arc plus the chord, the chord itself being twice r times the sine of half the angle.
This definition is part of the Circular Measure chapter in Cambridge O Level Additional Mathematics.
Circular Segment in context
A circular segment is the region cut off by a chord, found by subtracting from a sector the triangle on its two radii: \(A=\tfrac12r^2(\theta-\sin\theta)\), with \(\theta\) in radians throughout, including inside the sine. Minor and major regions on the same chord or pair of radii always add to the whole circle, so a major sector or segment is usually found by subtracting the minor one from \(\pi r^2\) rather than substituting the reflex angle directly. A perimeter is traced around the exposed boundary of a region, not quoted from a formula: an ordinary sector's perimeter is \(2r+r\theta\), but a shared internal edge inside a compound figure is never part of it.