A shaded region is usually one shape taken away from another, so the method is to name the pieces first. A segment is a sector minus a triangle, and a band between two radii is a large sector minus a small one.
This lesson follows arc length and sector area and is part of circular measure.
How do you split a shaded region?
Sketch the figure and label each piece. Decide whether the shaded area is a sum or a difference of the pieces, then find each piece with its own formula.
Worked example 1: a segment
A sector OAB has radius 10 cm and angle 1.2 radians. Find the area of the shaded segment between the chord AB and the arc, and its perimeter.
- Sector: ½ × 100 × 1.2 = 60 cm².
- Triangle OAB: ½ × 100 × sin 1.2 = 50 × 0.9320 = 46.60 cm².
- Segment: 60 − 46.60 = 13.40 cm².
Check with the direct formula: ½r²(θ − sinθ) = 50 × (1.2 − 0.9320) = 50 × 0.2680 = 13.40 cm².
For the perimeter, the arc is 10 × 1.2 = 12 cm. The chord is 2 × 10 × sin 0.6 = 11.29 cm. The perimeter is 12 + 11.29 = 23.29 cm.
Worked example 2: a band between two radii
Two sectors share centre O and angle 0.8 radians. The inner radius is 6 cm and the outer radius is 9 cm.
Area of the band: ½(9² − 6²) × 0.8 = ½ × 45 × 0.8 = 18 cm².
The mistake that costs marks
A common slip is to forget the triangle and call the sector the segment.
| Step | Wrong | Right |
|---|---|---|
| Region | Sector only | Sector minus triangle |
| Working | ½ × 100 × 1.2 | 60 − 46.60 |
| Area | 60 cm² | 13.40 cm² |
| Size check | Bigger than the shaded part in the sketch | Small, like the sliver drawn |
The segment in the sketch looks like a thin slice, so 60 cm² cannot be right. Comparing the answer with the sketch is the quickest check.
Check yourself
A sector has radius 12 cm and angle π/3. Find the area of the segment to 2 decimal places.
Answer
Area = ½r²(θ − sinθ) = 72 × (π/3 − sin(π/3)).
π/3 = 1.0472 and sin(π/3) = 0.8660, so the bracket is 0.1812.
Area = 72 × 0.18117 = 13.04 cm² (to 2 decimal places).
What to study next
The final lesson, circular-measure applications with consistent units, adds real-life contexts. Then test the chapter with the circular measure practice set.
The word-problem structure worksheet helps you split a figure into pieces. A teacher can guide the split in online one-to-one Additional Mathematics tuition.