A sector is a fraction of a full circle, and that fraction is always theta ÷ 360. Multiply this fraction by the circumference 2πr to get arc length, or by the area πr² to get sector area. The one step students forget: sector perimeter is not just the arc, it is the arc plus the two straight radii that close off the slice.
Arc length and sector area are two of the most formulaic topics in the whole syllabus: there are only two formulas, and both come from the same idea (a sector is a slice of the circle, so scale the circle's circumference or area by the slice's fraction). Students still lose marks, not because the formulas are hard, but because they mix up which formula gives a length and which gives an area, or because they answer "find the perimeter of the sector" with just the arc length and forget the two radii.
A second, smaller trap: O-Level E-Math works entirely in degrees. If you see a question written with radians, that belongs to A-Math, not E-Math. Every angle theta in this guide is in degrees, and the ÷360 in both formulas only works because a full turn is 360°.
Arc length
(θ ÷ 360) × 2πr
Sector area
(θ ÷ 360) × πr²
Sector perimeter
arc length + 2r
Segment area
sector area − triangle area
In every formula, θ is the angle at the centre in degrees, and r is the radius. Both the arc length and sector area formulas start from the same fraction θ ÷ 360, which is why it is worth writing that fraction down first, before you multiply it by anything.
A sector has radius r = 8 cm and angle θ = 135°. Find the arc length and the sector area, giving each answer in terms of π and then to 3 significant figures.
Solution: arc length
θ ÷ 360 = 135 ÷ 360.2πr = 2 × π × 8 = 16π.= (135 ÷ 360) × 16π = 6π cm (exact, in terms of π).6π = 18.8495... ≈ 18.8 cm.Solution: sector area
135 ÷ 360.πr² = π × 8² = 64π.= (135 ÷ 360) × 64π = 24π cm² (exact, in terms of π).24π = 75.3982... ≈ 75.4 cm².Quick check: the units tell you which formula to use. Arc length answers come out in cm (a length), sector area answers come out in cm² (an area). If your exact answer has π but you are not sure whether it should be a length or an area, check the units on the question first.
A sector has radius r = 10 cm and angle θ = 72°. Find the perimeter of the sector, to 3 significant figures.
Solution
= 72 ÷ 360, and 2πr = 2 × π × 10 = 20π.= (72 ÷ 360) × 20π = 4π cm = 12.566... cm.perimeter = arc length + 2r.perimeter = 12.566... + (2 × 10) = 12.566... + 20 = 32.566... cm.perimeter ≈ 32.6 cm.The step students get wrong
"Find the perimeter of the sector" is not the same question as "find the arc length". A sector's boundary has three parts: the curved arc, and two straight radii joining the centre to each end of the arc. Students who answer with the arc length alone (12.6 cm instead of 32.6 cm in the example above) lose the marks for the two radii, even though the arc length working was correct.
A segment is the region cut off by a chord: the curved part of the sector, minus the straight-sided triangle formed by the two radii and the chord. To find the area of a segment:
Segment area = sector area − triangle area
The triangle in a sector is isosceles, with two sides equal to r and the angle between them equal to θ, so its area uses ½r²sinθ, covered in the area of a triangle guide. Subtract this triangle area from the sector area (both calculated for the same r and θ) to get the segment area.
Calculator tip: make sure your calculator is in Degree mode (D on the display, not R) before finding sinθ for a segment area question. See the degree vs radian check guide if you are not sure how to confirm this on the fx-97SG.
A note on radians: A-Level and A-Math students meet an alternative pair of formulas, arc length = rθ and sector area = ½r²θ, where θ is measured in radians. These are not tested in O-Level E-Math (4052). Stick to the θ ÷ 360 versions above, with θ always in degrees.
How do I remember which formula is arc length and which is sector area?
Match the shape being scaled to the shape you want. Circumference 2πr is a length, so (θ ÷ 360) × 2πr gives a length, the arc. Circle area πr² is an area, so (θ ÷ 360) × πr² gives an area, the sector. Check the units in your final answer, cm for arc length, cm² for sector area, as a fast self-check.
Do I always need to find the arc length before the sector perimeter?
Yes. Sector perimeter is built from the arc length, so calculate the arc length first, then add 2r for the two radii. There is no shortcut that skips the arc length step.
What if the question gives the arc length and asks for the angle or the radius?
Use the same formula, arc length = (θ ÷ 360) × 2πr, and rearrange for whichever quantity is unknown. Substitute the known values first, then solve the equation for θ or for r, rather than trying to remember a separate rearranged formula.
Is the segment area formula on the exam formula sheet?
The MOE E-Math formula list gives arc length, sector area, and the triangle area formula ½ab sinC separately. You are expected to combine sector area and triangle area yourself to get segment area, so understanding the sector-minus-triangle idea matters more than memorising a combined formula. Questions often pair this with the circle theorems, which supply the angle at the centre before you can use it in either formula.
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