Arc: A Beginner’s Guide to Arc Length and Applications
What an arc is
An arc is a continuous portion of a curve between two points. In basic geometry it usually refers to part of a circle’s circumference, defined by its two endpoints and the central angle that subtends it.
Arc length (circle)
- Formula: For a circle of radius r and central angle θ (in radians),
Code
s = rθwhere s is the arc length.
- If the angle is given in degrees (θ°), convert to radians: θ (rad) = θ° × π/180. Then s = r × (θ° × π/180).
Arc length (general parametric curve)
For a smooth curve defined parametrically as x = x(t), y = y(t) for t in [a,b]:
- Arc length:
Code
s = ∫a^b sqrt((dx/dt)^2 + (dy/dt)^2) dt - For a function y = f(x) on [a,b]:
Code
s = ∫_a^b sqrt(1 + (f’(x))^2) dx
Worked example (circle)
- Radius r = 5, central angle 60°.
- Convert angle: 60° = π/3 rad.
- Arc length: s = 5 × π/3 = 5π/3 ≈ 5.236.
Applications
- Engineering: determining lengths of pipe/rail segments following curved paths.
- Architecture: measuring curved facades, arches, and structural elements.
- Computer graphics & animation: calculating motion along curved paths and stroke lengths.
- Navigation & geodesy: great-circle distances on Earth approximated as arcs on a sphere.
- Manufacturing/CNC: toolpaths for cutting/etching along curves.
Tips & common pitfalls
- Always use radians in the basic circle formula.
- For numerical integration of complex curves, subdivide the interval or use adaptive methods for accuracy.
- For arcs on ellipses or other conics there are no simple closed-form formulas; use elliptic integrals or numerical methods.
February 5, 2026
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