Library / Foundations: Maths / FMA-22
Foundations: Maths
Arc length and sector area (radians)
Arc length and sector area from radius and angle in radians.
- Ref
- FMA-22
- Rev
- v1
- Field
- Foundations: Maths
- Result
- s [m]
θ must be in radians. One radian is the angle for which the arc equals the radius; 2π radians make a full turn. 1 milliradian subtends 1 m at 1 km.
With the angle in radians, the arc length is just radius times angle.
Calculator
m
m
rad
- Sector area ½r²θ2.4 m²
- Angle in degrees68.75 °
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 (m) current value 2.4 m
rRadiusKey driver
0.660
θAngle
0.212
- The radian version is simpler because the 2π is absorbed into the unit.
- A milliradian (mil) is 1/1000 rad, so at range R an angle of θ mils spans R θ/1000.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| r Radius | 0.5 to 50 m | 0.6 | 60 | 59.4 | +10% | +10 % |
| θ Angle | 0.1 to 6 rad | 0.2 | 12 | 11.8 | +10% | +10 % |
Sweep of Angle
| Angle (rad) | Arc length (m) |
|---|---|
| 0.1 | 0.2 |
| 0.8375 | 1.675 |
| 1.575 | 3.15 |
| 2.313 | 4.625 |
| 3.05 | 6.1 |
| 3.788 | 7.575 |
| 4.525 | 9.05 |
| 5.263 | 10.53 |
| 6 | 12 |
Sweep one variable
Arc length (m)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
Angle
Angle at the centre in radians. Directly proportional.
Worked example with the values above
- Arc length: s = rθ = 2 × 1.2.
- Sector area: ½ × 2² × 1.2 = 2.4 m².
- Arc length 2.4 m.
Assumptions and limits
- θ is in radians.
- Circular arc.
Common mistakes
- Using degrees in this formula.
- Forgetting the ½ in the area.
Statistics and uncertainty
Angle measurement errors transfer directly to arc length in proportion; a 1 mrad angular error is 1 m at 1 km.
Where it is used
- Gunnery and sighting in mils.
- Rotating machinery arc distances.
Maths you need
- Radians: 2π rad = 360°
- Circles: radius and arc