§11.1 · Naming the regions
Sector vs. Segment
Drag point B around the circle to change ∠AOB. Switch the view to see the two radii carve out a sector, or the chord AB carve out a segment. The smaller piece is always called minor; the rest of the circle is the major one.
View
Instrument
Sector — the region enclosed by two radii and the arc between them. Segment — the region enclosed by a chord and the arc it cuts off. Unless a question says otherwise, "the sector" and "the segment" always mean the minor ones.
§11.1 · Formulas
Arc length and area of a sector
Change the radius and the angle θ. Both formulas come from the same idea — a sector of angle θ is just a fraction θ⁄360 of the whole circle.
Instrument
Length of arc = (θ⁄360) × 2πr · Area of sector = (θ⁄360) × πr²
§11.1 · Example 2 method
Area of a segment = sector − triangle
A segment is what's left of a sector once you cut away the triangle OAB. Adjust r and θ and watch the sector area, the triangle area, and their difference (the segment) all update — exactly the method used to solve Example 2 in the book.
Instrument
Drop a perpendicular OM from the centre to chord AB. It bisects AB and ∠AOB, giving two right triangles you can solve with sine and cosine. Once you know AB and OM, area of △OAB = ½ × AB × OM, and area of segment = area of sector − area of △OAB.
§11.1 · Note in the textbook
Minor + Major = the whole circle
Sweep θ and see the minor sector (bright) and major sector (dim) always add up to the full circle — and same for the two segments.
Instrument
Area of major sector = πr² − area of minor sector. Area of major segment = πr² − area of minor segment. The major sector's angle is always 360° − θ.
Applying the formulas
Real-world practice
These mirror three problems from the exercise — a grazing horse, a rotating clock hand, and umbrella ribs. Each is a sector in disguise.