Arc length is the radius times the angle — in radians
An arc is a piece of a circle's circumference. Its length depends on two things only: how far you are from the centre, and how much of a turn the arc covers. Put the angle in radians and the relationship is as simple as it gets: s = r θ. Nothing else appears, no π and no 360, because the radian is defined so that this is true.
That definition is worth stating plainly: one radian is the angle that subtends an arc equal in length to the radius. Sweep one radian on a 10-inch radius and you travel 10 inches along the curve. A full turn covers a circumference of 2πr, so a full turn is 2π radians ≈ 6.2832, and one radian is 180/π ≈ 57.2958°.
Working in degrees you must convert, either up front with θ = degrees × π/180 or by using the fraction-of-the-circle form s = 2πr × (θ°/360). Both give the same number; skipping the conversion and multiplying the radius by the degree count is the single most common error in this calculation, and it inflates the answer by a factor of 57.3. The degrees to radians calculator handles the conversion on its own.
This tool also reports the chord, the straight line between the arc's ends, because in practice you often measure one and need the other — a tape stretched across an opening reads the chord, while the material that has to wrap the curve is sized by the arc.
The three related quantities and why the chord is not proportional
Arc length is linear in the angle: double the sweep and you double the distance travelled, because you are covering twice as much of a circumference that is spread uniformly. Sector area is linear too, A = ½r²θ, which can also be written A = ½rs — the same shape as the triangle formula, with the arc playing the part of the base and the radius the height.
The chord behaves differently. Drop a perpendicular from the centre to the chord: it bisects both the chord and the central angle, leaving two right triangles with hypotenuse r and opposite angle θ/2. So chord = 2r·sin(θ/2), which is not proportional to θ. At small angles the two nearly agree — at 10° the chord is 99.87% of the arc — and they diverge steadily until at 180° the chord is the diameter, 2r, while the arc is πr, so the arc is 57% longer. Past 180° the chord starts shrinking again while the arc keeps growing, and at 360° the chord is zero.
Rearranging s = rθ gives the other two problems. θ = s/r recovers the angle from a measured arc, which is how you find the bend angle in an existing pipe or duct. r = s/θ recovers the radius, which is how you find the centreline radius a former was rolled to. Both need consistent units on s and r; the angle comes out in radians and this calculator converts it to degrees for you.
One practical caution on bends: in pipe and sheet-metal work, the arc that matters is the one on the neutral axis, not the centreline of the tube and not the outside surface. Which radius that corresponds to is a material and process question, not a geometric one.
Worked example: developed length of a 60° bend
A duct turns through 60° on a 10-inch centreline radius. How much material does the curve consume, and how far apart are the two ends?
- Convert the angle. θ = 60 × π/180 = 60 × 0.0174533 = 1.0471976 rad.
- Arc length. s = rθ = 10 × 1.0471976 = 10.4720 in. That is the developed length to add to the straight runs.
- Check the other way. s = 2πr × (60/360) = 62.8319 × 0.16667 = 10.4720 in. Agreed.
- Chord. 2r·sin(θ/2) = 20 × sin 30° = 20 × 0.5 = 10.0000 in exactly. The 60° case is the one where the chord equals the radius, because the two radii and the chord form an equilateral triangle.
- Sector area. A = ½r²θ = ½ × 100 × 1.0471976 = 52.3599 in², or equivalently ½rs = ½ × 10 × 10.4720.
- Arc against chord. 10.4720/10.0000 = 1.0472, so the curve is 4.72% longer than the straight line across it.
Now the reverse problem. You measure 15.0000 in along an existing curve of 10 in radius and want its included angle: θ = s/r = 15/10 = 1.5000 rad, which is 1.5 × 180/π = 85.9437°. And if you know an arc of 7.853982 in turns exactly 90°, the radius it was rolled to is r = s/θ = 7.853982/1.5707963 = 5.0000 in.
Reading arc, chord and sector against each other
Use the arc for anything that follows the curve: trim on a curved wall, gasket or seal length, belt wrap, conveyor contact, the developed blank for a rolled part, the distance a wheel travels for a given rotation. Use the chord for anything that spans it: a tape across an opening, the layout offset between two stakes, a straight brace inside a curve. Use the sector area for material coverage and for hydraulics, where a partially full round pipe's flow area starts from a sector.
The ratio of arc to chord is a compact measure of how sharp a curve is. Below about 30° the two agree within 1.2%, which is why short curves can be laid out with straight chords with almost no error — the basis of the chord-offset method used to stake highway curves. By 90° the arc is 11.1% longer than the chord, and by 180° it is 57.1% longer. If you have been substituting a chord for an arc, that table of ratios tells you what it cost you.
On angle unit: if a result looks about 57 times too large or too small, you have mixed degrees and radians. The number 57.2958 is the tell. And if the angle exceeds 360° the arc length keeps accumulating correctly — useful for spiral wraps and for cable on a drum — but the chord stops being meaningful as a single sweep, and this calculator says so.
Arc, chord and sector for a 10-unit radius
| Angle | Radians | Arc length | Chord | Arc ÷ chord | Sector area |
|---|---|---|---|---|---|
| 10° | 0.174533 | 1.74533 | 1.74311 | 1.0013 | 8.7266 |
| 30° | 0.523599 | 5.23599 | 5.17638 | 1.0115 | 26.1799 |
| 45° | 0.785398 | 7.85398 | 7.65367 | 1.0262 | 39.2699 |
| 60° | 1.047198 | 10.47198 | 10.00000 | 1.0472 | 52.3599 |
| 90° | 1.570796 | 15.70796 | 14.14214 | 1.1107 | 78.5398 |
| 120° | 2.094395 | 20.94395 | 17.32051 | 1.2092 | 104.7198 |
| 180° | 3.141593 | 31.41593 | 20.00000 | 1.5708 | 157.0796 |
| 360° | 6.283185 | 62.83185 | 0.00000 | — | 314.1593 |
The arc-to-chord ratio is θ ÷ (2 sin(θ/2)) and does not depend on the radius, so this column applies to a curve of any size. At 360° the chord is zero and the ratio is undefined.
Where arc length calculations go wrong
- Multiplying the radius by the angle in degrees. s = rθ needs radians. Doing it in degrees overstates the arc by 180/π = 57.2958 times.
- Substituting the chord for the arc. They differ by less than 1% only below about 28°. At 90° the gap is over 11%, which on a 10-foot radius is more than 1.5 feet of material.
- Using the outside radius on a bend. Developed length follows the neutral axis. Taking the outside surface of a thick-walled pipe overstates the blank.
- Confusing the central angle with an inscribed angle. An inscribed angle on the same arc is exactly half the central angle, so using it directly halves the arc.
- Mixing units between arc and radius. θ = s/r is only dimensionless if s and r are in the same unit; inches over feet gives an angle twelve times too large.
- Rounding π/180 too early. Carry at least seven digits on the conversion factor; rounding it to 0.0175 costs 0.0175/0.0174533 − 1 = 0.27% on every result.
Related circle quantities and where they take over
At θ = 360° the arc becomes the whole circumference, which is what the circumference calculator computes directly, and the sector becomes the whole disc, handled by the circle area calculator. Those are the boundary cases of the two formulas here.
Two related regions are often confused with the sector. A sector is the pie slice bounded by the arc and two radii; a segment is the smaller region between the arc and its chord, whose area is the sector minus the triangle, ½r²(θ − sin θ). If you need the second, use the circular segment area calculator; for the first on its own, the circular sector area calculator is the direct tool. The chord alone, including the case where you know the distance from the centre rather than the angle, is covered by the chord length calculator.
The same s = rθ relationship reappears far from geometry. Angular velocity times radius gives linear speed, v = ωr, which is the derivative of s = rθ with respect to time; a wheel of radius r turning through θ rolls exactly rθ along the ground without slipping. That identity is why odometers, belt drives, capstans and rotary encoders all reduce to arc length once you have the radius. For converting between the two angle measures on their own, use the radians to degrees calculator.