For a sector with radius r and angle θ in radians, the arc length is s = rθ and the area is A = ½r²θ. Each formula needs θ in radians, so convert first if the angle is in degrees.
This lesson follows converting degrees and radians and belongs to circular measure.
How do you use both formulas together?
Write down r and θ first, in radians. Then pick the formula for the quantity you want, and keep the units with the answer.
Worked example: one sector, three answers
A sector has radius 8 cm and angle 1.2 radians.
- Arc length: s = rθ = 8 × 1.2 = 9.6 cm.
- Area: A = ½r²θ = ½ × 64 × 1.2 = 38.4 cm².
- Perimeter: arc plus two radii = 9.6 + 2 × 8 = 25.6 cm.
Now reverse it. A sector with radius 6 cm has an arc of 15 cm. Then θ = s ÷ r = 15 ÷ 6 = 2.5 radians, and the area is ½ × 36 × 2.5 = 45 cm².
The mistake that costs marks
A common slip is to put a degree angle straight into the formula. Take a sector with r = 10 cm and angle 60°.
| Step | Wrong | Right |
|---|---|---|
| Angle used | 60 | π/3 = 1.047 radians |
| Arc length | 10 × 60 = 600 cm | 10 × 1.047 = 10.47 cm |
| Area | ½ × 100 × 60 = 3 000 cm² | ½ × 100 × 1.047 = 52.36 cm² |
| Size check | 600 cm is almost 10 times the circumference | Smaller than the circumference of 62.8 cm |
The circumference of the circle is 2π × 10 = 62.8 cm, so an arc of 600 cm cannot be part of it. The habit is to convert, then compare with the full circle.
Check yourself
A sector has radius 5 cm and area 20 cm². Find the angle in radians, the arc length and the perimeter.
Answer
From A = ½r²θ: 20 = ½ × 25 × θ, so θ = 40 ÷ 25 = 1.6 radians.
Arc length: s = 5 × 1.6 = 8 cm.
Perimeter: 8 + 2 × 5 = 18 cm.
Check the area: ½ × 25 × 1.6 = 20, which matches.
What to study next
Go on to finding shaded regions involving sectors, which combines sectors with triangles. Then use the circular measure practice set.
The word-problem structure worksheet helps you list what is given, and a mistake log shows whether you forget the two radii. To work with a teacher, see online one-to-one Additional Mathematics tuition.