Library / Foundations: Maths / FMA-09
Foundations: Maths
Arc length and sector area (degrees)
Part of a circle from the angle at its centre.
- Ref
- FMA-09
- Rev
- v1
- Field
- Foundations: Maths
- Result
- s [cm]
θ is the sector angle in degrees. The sector area is (θ/360) πr².
A sector is a fraction θ/360 of the whole circle, so its arc and area are the same fraction of the circumference and area.
Calculator
cm
cm
°
- Sector area18.85 cm²
- Angle in radians1.047 rad
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
Arc length (cm) current value 6.283 cm
rRadiusKey driver
1.0552.4
θSector angle
1.0531.4
- The angle must be in degrees here; A-level work uses radians.
- Arc length scales with r, sector area with r².
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| r Radius | 1 to 50 cm | 1.047 | 52.36 | 51.31 | +10% | +10 % |
| θ Sector angle | 10 to 300 ° | 1.047 | 31.42 | 30.37 | +10% | +10 % |
Sweep of Sector angle
| Sector angle (°) | Arc length (cm) |
|---|---|
| 10 | 1.047 |
| 46.25 | 4.843 |
| 82.5 | 8.639 |
| 118.8 | 12.44 |
| 155 | 16.23 |
| 191.3 | 20.03 |
| 227.5 | 23.82 |
| 263.8 | 27.62 |
| 300 | 31.42 |
Sweep one variable
Arc length (cm)Hover, touch, or use the arrow keys on the chart
Variables and how each one contributes
s
Arc length
What the equation gives you. Everything else feeds it.
r
Radius
Radius of the circle. Directly proportional.
θ
Sector angle
Angle at the centre. Directly proportional.
Worked example with the values above
- Fraction of the circle: 60/360 = 0.1667.
- Whole circumference: 2π × 6 = 37.7 cm.
- Arc = 6.283 cm.
Assumptions and limits
- θ measured at the centre.
- A true circle.
Common mistakes
- Using radians in a degrees formula.
- Forgetting to divide by 360.
Statistics and uncertainty
Angle readings with a protractor are good to about ±1°, which is ±1.7% on a 60° sector.
Where it is used
- Radar sweep and sector coverage.
- Curved path lengths.
Maths you need
- Circles: circumference and area
- Fractions: θ/360