Arc Length Calculator
Enter a circle’s radius and a central angle and the tool returns the arc length along with the chord (straight-line distance) and sector area. Supports input in degrees or radians. Useful for woodworking templates, stair and ramp layout, road design and any geometry homework involving curved sections.
How to calculate an arc length
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1
Enter the radius
The distance from the center of the circle to the arc. Any length unit works — output matches the input unit.
-
2
Pick angle unit
Degrees (0-360) or radians (0-2π). Most manual work uses degrees; calculus and physics use radians.
-
3
Enter the central angle
The angle subtended at the center of the circle between the two radii that bound the arc.
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4
Read arc length, chord and area
Arc length is the distance along the curve. Chord is the straight line between endpoints. Sector area is the pie-slice area between the two radii and the arc.
The arc length formula
In radians:
arc_length = r × θ
Where r is the radius and θ is the central angle in radians. This is why radians exist as a unit — the formula is clean. In degrees you have to convert:
arc_length = r × (θ_degrees × π / 180)
Related formulas for the same setup
- Chord length:
2 × r × sin(θ / 2) - Sector area:
0.5 × r² × θ(θ in radians) - Segment area (between chord and arc):
0.5 × r² × (θ - sin(θ)) - Arc height (sagitta):
r - r × cos(θ / 2)— how far the arc rises above the chord midpoint.
Common practical cases
| Scenario | Typical angle | Notes |
|---|---|---|
| Curved wooden baseboard | 15-45° | Chord ≈ arc for small angles |
| Staircase tread on a circular stair | 20-30° | Angle × number of treads = 360° |
| Road curve (horizontal alignment) | 5-45° | Design speed drives maximum curvature |
| Pipe bend | 22.5° / 45° / 90° | Standard fitting angles |
| Archway | 60-180° | Semicircle is 180° |
Small-angle approximation
For angles under about 15° (0.26 rad), arc length and chord are within 1% of each other. For very small angles you can treat the curve as a straight line without significant error — useful for segmenting a curve into short straight panels (e.g. for CNC or laser cutting with polyline approximations).
Radians vs. degrees: which to use
| Context | Preferred unit |
|---|---|
| Everyday geometry | Degrees |
| Physics, calculus | Radians |
| Navigation, surveying | Degrees (often with minutes and seconds) |
| Programming (math libs) | Radians (almost universal) |
If your code is calling Math.sin(x) or sin(x) without a conversion, x is in radians. Converting degrees to radians: multiply by π/180 ≈ 0.01745.
Frequently Asked Questions
Arc length is a fraction of the circumference, proportional to the central angle. A 90° arc is one quarter of the full circumference (2πr). Arc length = (angle / 360°) × 2πr in degrees, or r × θ in radians.
Rearrange: r = arc_length / θ (θ in radians), or r = arc_length × 180 / (π × θ_degrees). Useful when you know the curve length and sweep but need to back out the radius.
The arc wraps past the full circle. Mathematically the formula still works (e.g. 720° gives 2 × circumference) but physically an arc of more than 360° is just the circle traced multiple times. For practical work, reduce modulo 360°.
A helical stair has each step subtending a central angle. Tread arc length at the outer edge = radius × per-step angle. The walking line (typically 12-18 inches from the inner edge) uses a smaller radius and shorter arc, which is why building codes specify minimum walking-line tread depth rather than minimum total tread.
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