Variables
| Symbol | Quantity | Unit |
|---|---|---|
| L | Arc length | |
| r | Radius of the circle | |
| theta | Central angle | radians |
What it means
The arc length of a sector of a circle is directly proportional to the radius and the angle subtended at the centre, provided the angle is in radians. The formula L = rθ is derived from the definition of radian. This is used in many practical applications: in manufacturing to calculate the length of a curved part, in physics for circular motion (path length = radius × angular displacement), in surveying, and in computer graphics for curved paths. The formula also generalises to any parametric curve through integration. Understanding this relation is fundamental for geometry, trigonometry, and engineering design involving curved shapes.
Worked example
Arc Length – Two Examples
Real‑World| Parameter | Value |
|---|---|
| r | 100 m |
| θ | 0.5 rad |
| Parameter | Value |
|---|---|
| r | 0.05 m |
| θ | 1.2 rad |
Common mistakes
- Arc length: L = r·θ – θ must be in radians, not degrees.
- Units: r and L have the same units (e.g., metres); θ is dimensionless.
- Sign: θ can be positive (counterclockwise) or negative (clockwise) – length is positive using absolute value.
- Full circle: For a full revolution (θ = 2π), L = 2πr (circumference).
- Sector area: Not to be confused with sector area (A = ½r²θ) – different formula.
Applications
The arc length of a circular arc, L = r·θ, relates the radius and the central angle (in radians) to the length of the curved segment. This formula is fundamental in mechanical engineering for designing gears, belts, and pulleys, as well as in civil engineering for road curves and railway tracks. In physics, it calculates the distance travelled by a point on a rotating wheel. In navigation, it is used for great‑circle distances. The formula also appears in computer graphics for rendering circles and arcs. By using radians, the calculation becomes straightforward, avoiding the need for conversion. This formula is a cornerstone of circular geometry and rotational dynamics.
- Design of curved roads, railways, and pipelines
- Gear and belt drive geometry in mechanical systems
- Rotational motion analysis (distance travelled on a wheel)
- Navigation and geodesy – great‑circle arcs on the Earth's surface
- Computer graphics and CAD modelling of circular shapes
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