A sector is a slice of a circle. Arc length is the distance along the curved edge, and sector area is the space inside the slice. Both take the fraction θ/360 of the whole circle, but they start from different whole-circle values: circumference 2πr for the arc, area πr² for the sector.
This lesson sits in area, perimeter and surface area. It builds on decomposing a compound shape, since sectors appear as part of larger figures.
How do you keep the two formulas apart?
Ask what the answer is measuring. A distance along a curve is a length, so it uses the circumference formula and its unit has no square. A region is an area, so it uses πr² and its unit is squared.
- Arc length = θ/360 × 2πr
- Sector area = θ/360 × πr²
- Sector perimeter = arc length + 2r
The fraction θ/360 is the same in all three. Only the whole-circle quantity changes.
Step by step
- Read what is wanted: length, area or perimeter.
- Write the fraction θ/360 and simplify it if you can.
- Write the corresponding whole-circle value: 2πr or πr².
- Multiply and keep the full calculator value.
- For a perimeter, add the two radii to the arc.
- Round at the end and check the unit.
Worked example
A sector has radius 9 cm and angle 140°. Find the arc length, the area and the perimeter.
Fraction: 140/360 = 7/18.
Arc length: 7/18 × 2 × π × 9 = 7/18 × 56.55 = 21.99, so 22.0 cm (3 s.f.).
Sector area: 7/18 × π × 9² = 7/18 × 254.47 = 98.96, so 99.0 cm² (3 s.f.).
Perimeter: arc + two radii = 21.99 + 9 + 9 = 39.99, so 40.0 cm (3 s.f.).
Check. 140° is a little under half a circle. Half the circumference is about 28.3 cm, and the arc is 22.0 cm, which is smaller, as expected.
The mistake to watch for
A student uses the area formula when the question asks for the curved length.
Mistaken answer: arc length = 140/360 × π × 9² = 99.0 cm
This is the sector area with a length unit attached, so the unit already reveals the error.
The correction is to use 2πr, the circumference, for any curved length. Another slip is to give 21.99 cm as the perimeter and forget the two straight edges.
Check yourself
1. A sector has radius 6 cm and angle 60°. Find its arc length and its area.
Show answer
Fraction = 60/360 = 1/6. Arc = 1/6 × 2π × 6 = 2π = 6.283, so 6.28 cm. Area = 1/6 × π × 36 = 6π = 18.85, so 18.8 cm².
2. A sector of radius 10 cm has an arc length of 5π cm. Find the angle.
Show answer
5π = θ/360 × 2π × 10 = θ/360 × 20π. Divide both sides by π: 5 = θ × 20/360, so θ = 5 × 360 ÷ 20 = 90°.
3. Find the perimeter of a quarter circle of radius 8 cm.
Show answer
Arc = ¼ × 2π × 8 = 4π = 12.57. Add two radii: 12.57 + 16 = 28.57, so 28.6 cm.
Where this leads next
Move from flat curved shapes to solids with unfolding a prism to find surface area. Try sector questions in the mixed practice set, and use the non-calculator working trainer to handle answers left in terms of π.
If you can do each formula but choose the wrong one under pressure, a teacher can ask you why you chose it and help you correct the reasoning. This is part of online one-to-one Mathematics tuition.