Formula sheet
Arc length
(θ/360°) × 2πr is given on the exam formula sheet. You must still know it is a fraction of the full circumference, that θ is measured in degrees at the centre, and to keep the ×2πr (circumference) part, not πr².
What the symbols mean
- θ Angle at the centre subtended by the arc, in degrees.
- r Radius of the circle (length unit).
- π Pi ≈ 3.142 or 22/7.
Given in the exam, or memorise?
(θ/360°) × 2πr is given on the exam formula sheet. You must still know it is a fraction of the full circumference, that θ is measured in degrees at the centre, and to keep the ×2πr (circumference) part, not πr².
Why it works
An arc is just a fraction of the whole circumference.
- A full turn is 360°, and its length is the whole circumference 2πr.
- An angle θ covers the fraction θ/360° of that turn.
- So arc length = (θ/360°) × 2πr.
Worked example 1
Find the length of an arc that subtends 90° at the centre of a circle of radius 14 cm. (Use π = 22/7)
- Arc = (θ/360°) × 2πr
- = (90/360) × 2 × 22/7 × 14
- 2 × 22/7 × 14 = 88 (full circumference)
- = (1/4) × 88 = 22
Worked example 2
An arc of length 11 cm is drawn in a circle of radius 21 cm. Find the angle at the centre.
(Use π = 22/7)
- Arc = (θ/360°) × 2πr
- 2πr = 2 × 22/7 × 21 = 132
- 11 = (θ/360) × 132
- θ/360 = 11 ÷ 132 = 1/12
- θ = 360 ÷ 12 = 30
Where students go wrong
- Using πr² (area) instead of 2πr (circumference) inside the fraction.
- Forgetting to divide by 360°, so you get the whole circumference.
- Reading θ in a wrong unit, it must be in degrees here.
- Mixing radius and diameter in the 2πr part.
Use it with
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
What is the difference between arc length and area of a sector?
Arc length is a distance along the curved edge, so it uses the circumference part 2πr and comes out in cm. Area of a sector is the pie-slice region, so it uses πr² and comes out in cm².
Both take the same fraction θ/360°; the difference is whether you multiply by 2πr or by πr².
Do I need radians for arc length in SPM?
No. The SPM 1449 formula uses degrees: (θ/360°) × 2πr.
Keep θ in degrees and divide by 360°. Radians belong to higher-level courses.
As long as you read the centre angle in degrees and place it over 360, the formula works directly.
How do I find the angle if I know the arc length?
Set the arc length equal to (θ/360°) × 2πr, work out the full circumference 2πr, then divide the arc by it to get the fraction θ/360°. Multiply that fraction by 360° to get θ.
It is the same formula rearranged, you are just solving for θ instead of the arc.