Sector Area

For radius $r$ and angle $\theta$ in radians:

$A = \tfrac{1}{2}r^2\theta$

Same family as arc length, same radians requirement, same multidirectional use, plus one derived result that decides most of the topic’s hard marks.

The segment: the formula worth deriving

The segment (between a chord and its arc) is a sector minus the isosceles triangle on the same angle:

Area of segment $= \tfrac{1}{2}r^2\theta - \tfrac{1}{2}r^2 \sin \theta =$ $\tfrac{1}{2}r^2(\theta - \sin \theta)$

The triangle term comes from $\tfrac{1}{2}ab \sin C$ with both sides $r$, the bridge between this topic and trigonometry. Write the decomposition (sector $-$ triangle) rather than quoting the result: the decomposition line is the method mark, and it survives a formula blank.

$r = 10$, $\theta = 1.4$: segment area $= \tfrac{1}{2} \times 100 \times (1.4 - \sin 1.4) \approx 50 \times 0.4146 \approx 20.7$ cm$^2$ Calculator in radian mode for $\sin 1.4$, degree mode gives a plausible wrong number.

Shaded regions: name the pieces

0606’s favourite format is a composite diagram, overlapping sectors, a sector with a triangle removed, two circles touching. The universal method: decompose into named pieces (sector $OAB$, triangle $OAB$, segment, …), compute each with its own $r$ and $\theta$, then add/subtract per the shading. Annotate the diagram; the labelled decomposition is visible method even before any number appears. The hardest variants make you find $\theta$ or $r$ first, from a right triangle in the figure, an arc condition, or an isosceles geometry, so expect a trig step before the area step.

Reverse problems appear too: “the sector’s area is 75 cm$^2$ and its radius 10 cm” $\to$ $\theta = \frac{2A}{r^2} = 1.5$, output already in radians, no conversion.

Exactness

Paper 1 versions cooperate: $r = 6$, $\theta = \frac{\pi}{3}$ $\to$ $A = 6\pi$ exactly. Keep $\pi$ symbolic; keep $\sin \theta$ exact when $\theta$ is a standard angle.

Common mistakes

• Degrees in $\tfrac{1}{2}r^2\theta$, or degree-mode $\sin \theta$ alongside radian $\theta$
• Segment computed as sector $-$ wrong triangle (the triangle is $\tfrac{1}{2}r^2 \sin \theta$, two sides $r$, included angle $\theta$)
• The $\tfrac{1}{2}$ dropped in either formula
• Shaded regions attacked without naming pieces, unfollowable working, unawardable marks
• Intermediate values over-rounded before the final subtraction (carry precision)

Full topic context: Circular Measure notes.