Arc Length Calculator
Find the arc length of a circular arc from any two known values: radius (or diameter) and central angle, arc length and angle, or arc length and radius. Switch the angle unit between degrees and radians, and choose metric or imperial output. Every result comes with a step-by-step breakdown and the full geometry of the arc slice.
Formula
Worked example
Radius 5, central angle 60 degrees. Step 1: convert to radians, theta = 60 times pi over 180 = pi/3 approximately 1.0472 rad. Step 2: arc = 5 times 1.0472 = 5.2360 units. Step 3: chord = 2 times 5 times sin(pi/6) = 5.0000 units. Step 4: sagitta = 5 times (1 minus cos(pi/6)) approximately 0.6699 units. Step 5: sector area = 0.5 times 25 times 1.0472 = 13.0900 units squared.
How to calculate arc length
An arc is the curved portion of a circle between two points, and its length depends on just two things: the radius and the central angle. The core formula is s = r times theta, where theta is the central angle in radians. Because degrees are more intuitive for most people, the calculator automatically converts: theta = (alpha times pi) divided by 180. So for a 90-degree arc on a circle of radius 10, theta = pi over 2 and the arc is 10 times pi over 2, roughly 15.708 units. The formula works in any consistent unit: metres, centimetres, inches, or feet.
Reverse solving: find radius or central angle
Sometimes you know the arc length and need to find a missing quantity. If you have the arc length and the central angle, the radius is simply r = s divided by theta. If you have the arc length and the radius, the central angle in radians is theta = s divided by r, which you can convert to degrees by multiplying by 180 over pi. This calculator handles both reverse modes automatically: choose "Solve for: Radius" or "Solve for: Central angle" from the dropdown and enter the two known values.
Diameter as an alternative input
If you know the diameter rather than the radius, enter it in the diameter field and leave the radius blank. The calculator treats diameter = 2 times radius, so the rest of the computation is identical. This is handy when working with pipes, wheels, or round objects where the across measurement is easier to take with a ruler.
Chord, sector area, sagitta, and sector perimeter
Four related quantities appear alongside the arc. The chord is the straight line connecting the arc endpoints: c = 2r times sin(theta over 2). It is always shorter than the arc. The sector area is the pie-slice region bounded by the arc and the two radii: A = one-half times r-squared times theta. The sagitta (from Latin for arrow) is the perpendicular height from the chord to the arc midpoint: h = r times (1 minus cos(theta over 2)). A large sagitta means a deeply curved arc; a sagitta close to zero means a nearly flat arc. The sector perimeter is the total boundary of the slice: arc length plus two radii.
Degrees versus radians
Degrees divide a full rotation into 360 equal parts, a tradition from Babylonian astronomy. Radians are the mathematically natural unit: one radian is the angle for which the arc length equals the radius, making a full circle exactly 2 pi radians. The compact formula s = r times theta only works with theta in radians. Plug a degree value into that formula without converting and you will overstate the arc by a factor of about 57.3 (the number of degrees per radian). This calculator accepts both: pick the unit that matches your source data and it converts automatically.
Arc geometry for common central angles (radius = r)
| Angle | Radians | % of circle | Arc length | Chord | Sagitta | Sector area |
|---|---|---|---|---|---|---|
| 30° | pi/6 = 0.5236 | 8.33% | 0.5236 r | 0.5176 r | 0.0670 r | 0.2618 r² |
| 45° | pi/4 = 0.7854 | 12.50% | 0.7854 r | 0.7654 r | 0.1522 r | 0.3927 r² |
| 60° | pi/3 = 1.0472 | 16.67% | 1.0472 r | 1.0000 r | 0.2679 r | 0.5236 r² |
| 90° | pi/2 = 1.5708 | 25.00% | 1.5708 r | 1.4142 r | 0.5858 r | 0.7854 r² |
| 120° | 2pi/3 = 2.0944 | 33.33% | 2.0944 r | 1.7321 r | 1.0000 r | 1.0472 r² |
| 180° | pi = 3.1416 | 50.00% | 3.1416 r | 2.0000 r | 2.0000 r | 1.5708 r² |
| 270° | 3pi/2 = 4.7124 | 75.00% | 4.7124 r | 1.4142 r | 2.5858 r | 2.3562 r² |
| 360° | 2pi = 6.2832 | 100.00% | 6.2832 r | 0 r | 2 r | 3.1416 r² |
All lengths are multiples of the radius r. Sector area is in r-squared units.
Frequently asked questions
What is the formula for arc length?
Arc length s = r times theta, where r is the radius and theta is the central angle in radians. In degrees: s = 2 times pi times r times (angle divided by 360). Both give the same result. This calculator converts your degree input to radians automatically, so either angle unit works.
How do I find the arc length from the diameter?
If you know the diameter d rather than the radius, just use r = d divided by 2. Then arc length = (d over 2) times theta. In this calculator, enter the diameter in the diameter field and the central angle, and it does the conversion for you.
Can I solve for the radius or angle instead of arc length?
Yes. If you know the arc length and the central angle, the radius is r = s divided by theta (with theta in radians). If you know the arc length and radius, the central angle in radians is theta = s divided by r. Use the "Solve for" dropdown to pick which quantity to find, then enter the two known values.
What is the sagitta of an arc?
The sagitta (Latin for arrow) is the perpendicular distance from the midpoint of the chord to the midpoint of the arc. It measures how much the arc bulges above its chord. The formula is h = r times (1 minus cos(theta over 2)). For a very small angle the sagitta is nearly zero; for a semicircle (180 degrees) the sagitta equals the radius.
How is arc length different from chord length?
The arc follows the curved edge of the circle; the chord is the straight line joining the same two endpoints. The chord is always shorter than or equal to the arc. They are equal only for a zero-degree angle (both zero). At 180 degrees the chord equals the diameter while the arc is pi times the radius, roughly 57% longer.
What is the sector perimeter?
The sector perimeter is the total boundary length of a pie-slice region: it is the arc length plus the two straight radii that form the sides of the slice. So sector perimeter = arc + 2r. This is different from the sector area, which is the area of that region.