Arc Length Calculator
Calculate the exact length along any curve from its radius and central angle. Find how much curved edging, railing, or trim you need — and see how much longer the arc is than the straight chord.
units (m, ft, cm…)
Enter your values above to see the results.
Tips & Notes
- ✓Never order curved edging, railing, or trim material based on the straight-line chord measurement — always calculate the actual arc length. The difference ranges from 1% (shallow curves) to 57% (semicircles).
- ✓To measure the radius of an existing curve on site: measure the chord length (straight line between two endpoints on the curve), then measure the sagitta (height of the curve from the midpoint of the chord to the curve). Radius = (chord²/4 + sagitta²) / (2 × sagitta).
- ✓Convert degrees to radians before using the arc formula: multiply degrees by π/180 ≈ 0.01745. A 90° angle = 90 × 0.01745 = 1.5708 radians.
- ✓For curved concrete formwork, add 10-15% to the calculated arc length for overlap at stakes and the additional length needed to bend the form board around the curve.
- ✓When laying out a curved path or garden border on site, use a stake and string at the center point as a compass — walk the string around to mark the arc precisely at the calculated radius.
Common Mistakes
- ✗Using the chord length (straight line between endpoints) instead of the arc length to order curved materials — this is almost guaranteed to produce a shortage on curves greater than 30°.
- ✗Forgetting to convert degrees to radians before applying the arc formula — arc length = r × θ only works when θ is in radians, not degrees.
- ✗Measuring the radius to the wrong edge — for curved path edging, measure to the center of the edging material, not the inside or outside edge. For a 4-inch wide edging, inside radius and outside radius differ by 4 inches.
- ✗Treating a series of small straight sections as equivalent to a curve — approximating a curve with straight-line segments underestimates the total length. The finer the approximation, the closer to true arc length.
- ✗Confusing radius and diameter when entering values — using diameter where radius is expected produces an arc length twice as large as correct, causing a significant material over-order.
Arc Length Calculator Overview
Arc length calculation is needed whenever a curved section of any structure must be measured, cut, or estimated for materials — curved garden edging, arched door headers, circular staircase railings, road curves, and curved deck sections all require knowing the actual length along the curve rather than the straight-line chord distance between the two endpoints.
Arc length formula using central angle:
Arc Length = r × θ (θ must be in radians) | To convert degrees to radians: θ_rad = θ_deg × (π ÷ 180)
EX: Curved garden path — radius 20 ft, central angle 75° → θ = 75 × π/180 = 1.309 rad → Arc = 20 × 1.309 = 26.18 ft of edging material neededArc length formula using chord and radius (when angle is unknown):
Arc Length = 2r × arcsin(chord ÷ 2r)
EX: Arched window header — chord (straight span) 6 ft, radius of curvature 8 ft → Arc = 2 × 8 × arcsin(6/16) = 16 × arcsin(0.375) = 16 × 0.3844 rad = 6.15 ft of curved trim neededArc length reference by radius and angle:
| Radius | 45° Arc | 90° Arc | 120° Arc | 180° Arc (semicircle) | 360° (full circle) |
|---|---|---|---|---|---|
| 2 ft | 1.57 ft | 3.14 ft | 4.19 ft | 6.28 ft | 12.57 ft |
| 4 ft | 3.14 ft | 6.28 ft | 8.38 ft | 12.57 ft | 25.13 ft |
| 6 ft | 4.71 ft | 9.42 ft | 12.57 ft | 18.85 ft | 37.70 ft |
| 10 ft | 7.85 ft | 15.71 ft | 20.94 ft | 31.42 ft | 62.83 ft |
| 15 ft | 11.78 ft | 23.56 ft | 31.42 ft | 47.12 ft | 94.25 ft |
| 20 ft | 15.71 ft | 31.42 ft | 41.89 ft | 62.83 ft | 125.66 ft |
| Central Angle | Arc Length | Chord Length | Arc ÷ Chord Ratio | Practical Note |
|---|---|---|---|---|
| 30° | 0.5236r | 0.5176r | 1.012 | Nearly identical — flat curves |
| 60° | 1.0472r | 1.0000r | 1.047 | ~5% longer than chord |
| 90° | 1.5708r | 1.4142r | 1.111 | ~11% longer than chord |
| 120° | 2.0944r | 1.7321r | 1.209 | ~21% longer than chord |
| 180° | 3.1416r | 2.0000r | 1.571 | 57% longer — semicircle |
Frequently Asked Questions
Determine the radius of the curve and the central angle. Arc length = r × (angle in radians) = r × (degrees × π/180). Example: a 270° curved border with 8-ft radius: Arc = 8 × (270 × π/180) = 8 × 4.712 = 37.7 ft of edging. Add 5-10% for connections and overlaps. If you only know the straight-line span and how high the curve rises at its center, use the chord-sagitta method to find the radius first.
The chord is the straight-line distance between two points on a circle. The arc is the actual curved path between those same two points along the circle. Arc length is always longer than the chord. For shallow curves (central angle under 30°), the difference is small (under 1.5%). For a 90° arc, the arc is 11% longer than the chord. For a semicircle (180°), the arc is 57% longer. Always use arc length when ordering curved materials.
Measure the chord: a straight line between two points on the curve. Measure the sagitta: the perpendicular distance from the midpoint of the chord to the curve itself. Then: Radius = (chord²/4 + sagitta²) / (2 × sagitta). Example: chord = 10 ft, sagitta = 1.5 ft. Radius = (100/4 + 2.25) / (2 × 1.5) = (25 + 2.25) / 3 = 9.08 ft. This works for any visible curve without needing to know the center point.
Multiply degrees by π/180 (≈ 0.01745) to convert to radians. Multiply radians by 180/π (≈ 57.296) to convert to degrees. Key values to memorize: 30° = π/6 ≈ 0.5236 rad, 45° = π/4 ≈ 0.7854 rad, 60° = π/3 ≈ 1.0472 rad, 90° = π/2 ≈ 1.5708 rad, 180° = π ≈ 3.1416 rad, 360° = 2π ≈ 6.2832 rad.
Model the curve as a circular arc. For a road curve defined by radius R and deflection angle θ: Arc = R × θ (in radians). Highway engineers use degree of curve (D), where radius = 5729.58/D. Example: a 300-ft radius curve turning through 40°: Arc = 300 × (40 × π/180) = 300 × 0.6981 = 209.4 ft. Multiply by the road width to get paving area: 209.4 ft × 24 ft wide = 5,026 ft².
Yes, using two steps. First find the radius from the chord (c) and sagitta/rise (s): R = (c²/4 + s²) / (2s). Then find the central angle: θ = 2 × arcsin(c / (2R)). Finally: Arc = R × θ (with θ in radians). Example: chord 12 ft, rise 2 ft. R = (144/4 + 4) / (2×2) = (36+4)/4 = 10 ft. θ = 2 × arcsin(12/20) = 2 × arcsin(0.6) = 2 × 0.6435 = 1.287 rad. Arc = 10 × 1.287 = 12.87 ft.