Arc Length Calculator
The arc length calculator finds the distance along a circle’s curve between two points on it. Give it any two of the three values that define an arc, the radius, the central angle, and the arc length itself, and it solves for the third, in degrees or radians, with the working shown step by step. It also returns the companion measures for the same setup: the chord, the sector area, the segment area, and the sagitta.
The whole tool rests on one clean relationship: s = r x theta, where theta is the central angle in radians. That single formula is why radians exist. A central angle of 1 radian always cuts an arc exactly as long as the radius, so once the angle is in radians the multiplication is direct. Enter degrees instead and the calculator converts first (theta = degrees x pi / 180), then applies the same formula.
Circle mode uses exact formulas, the same ones printed under the result. Angles up to a full turn (360 degrees or 2 pi radians) are accepted.
Quick Answer
Key takeaways
- s = r x theta only works directly when theta is in radians. In degrees, convert first: theta = degrees x pi / 180.
- Any two of radius, angle, and arc length determine the third. This tool solves in all three directions.
- Arc length is not chord length: the chord cuts straight across and is always shorter (14.1421 vs 15.708 in the example).
- An arc over 180 degrees is a major arc; at or under 180 degrees it is a minor arc. The verdict badge under the result labels yours.
- Quick Answer
- The arc length formula, and why radians make it simple
- How to use the arc length calculator
- Worked example: radius 10, angle 90 degrees
- Arc, chord, sector, segment: four measures people mix up
- Reference table: common angles at radius 10
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
The arc length formula, and why radians make it simple
s = r x theta, with theta in radians. The formula is really the definition of a radian read backwards: one radian is the angle that cuts an arc equal to the radius, so an angle of theta radians cuts an arc theta times as long.
The degree form explains the intuition the radian form hides: an arc is a fraction of the whole circumference, and the fraction is theta / 360. A 90 degree arc is a quarter of the circle, a 60 degree arc is a sixth. Both forms are the same statement, which is why the calculator shows the radian conversion as its first step whenever you enter degrees. These are the standard definitions catalogued in the MathWorld arc length entry and used across geometry textbooks.
How to use the arc length calculator
Pick what you are solving for first, because it decides which fields you need. The steps below follow the worked example.
Choose Solve for. Arc length (you have radius and angle), Radius (you have arc and angle), or Central angle (you have arc and radius).
Match the angle unit to your problem. A protractor, drawing, or spec sheet almost always means degrees. A calculus or physics problem usually means radians. Entering a radian value as degrees is the single most common arc length error.
Read the steps, not just the answer. The result box prints the conversion and every companion formula with your numbers in it, so a homework answer can be checked line by line.
Worked example: radius 10, angle 90 degrees
The calculator is pre-filled with this case. Convert 90 degrees to radians: 90 x pi / 180 = 1.5708 rad. Then s = 10 x 1.5708 = 15.708.
| Quantity | Formula | Value |
|---|---|---|
| Arc length | s = r x theta | 15.708 |
| Chord length | 2 r sin(theta / 2) | 14.1421 |
| Sector area | (1/2) r^2 theta | 78.5398 |
| Segment area | (1/2) r^2 (theta – sin theta) | 28.5398 |
| Sagitta | r – r cos(theta / 2) | 2.9289 |
| Full circumference | 2 pi r | 62.8319 |
Run it backwards to test the other modes. Solve for Radius with arc 15.708 and angle 90 degrees and you get r = 15.708 / 1.5708 = 10. Solve for Central angle with radius 10 and arc 15.708 and you get theta = 1.5708 rad = 90 degrees. A special case worth knowing: the arc of a semicircle. A semicircle is a 180 degree arc, so its length is pi x r, half the circumference. That is the curved edge the area of a semi-circle page adds the diameter to when it computes a semicircle perimeter.
Arc, chord, sector, segment: four measures people mix up
They come from the same two inputs, which is exactly why they get confused. The arc is the curved distance. The chord is the straight line between the arc’s endpoints. The sector is the pie slice bounded by two radii and the arc. The segment is the smaller region between the chord and the arc: the sector minus its triangle.
Use the arc or chord when you need
- Material that follows the curve: trim, edging, belt and track lengths (arc).
- Clearance and span checks between two points on a curve (chord).
- Distance travelled by a point on a rotating wheel or arm (arc).
Use the sector or segment when you need
- Area swept by a wiper, sprinkler, or rotating arm (sector).
- Area of a circular offcut, lens cap, or part-filled round tank cross-section (segment).
- The sagitta when you need the height of the arc above its chord, for arch and camber work.
Arc length also shows up outside pure geometry. A point on a wheel that rolls without slipping travels one circumference per revolution, and partial turns are arc length problems; the same distance-over-time thinking drives the acceleration calculator when motion is measured along a path. Ellipses break the tidy pattern: an ellipse perimeter has no exact arc formula and needs an approximation, which is why the area of an oval calculator uses Ramanujan’s approximation for its perimeter instead of a closed form.
Reference table: common angles at radius 10
At a fixed radius, arc length scales in direct proportion to the angle. Doubling the angle doubles the arc, in either unit system.
| Angle | Angle (rad) | Arc length (r = 10) | Sector area | Chord |
|---|---|---|---|---|
| 30 degrees | 0.5236 | 5.236 | 26.1799 | 5.1764 |
| 45 degrees | 0.7854 | 7.854 | 39.2699 | 7.6537 |
| 60 degrees | 1.0472 | 10.472 | 52.3599 | 10 |
| 90 degrees | 1.5708 | 15.708 | 78.5398 | 14.1421 |
| 180 degrees | 3.1416 | 31.4159 | 157.0796 | 20 |
| 360 degrees | 6.2832 | 62.8319 | 314.1593 | 0 |
The 60 degree row is a good sanity check for any arc tool: the chord equals the radius exactly (both 10), because the two radii and the chord form an equilateral triangle. If a calculator returns anything else for that row, its angle handling is broken.
What competing calculators miss
Winning move 1: all three solve directions. Most arc tools only run forwards. Examples.com, for instance, accepts radians only and solves only for arc length. This tool also solves for radius and for angle, which is the form surveying, machining, and belt-length problems actually arrive in.
Winning move 2: steps you can hand in. Engineering tools such as FIRGELLI’s show formulas well, but the printed per-problem steps here (conversion, substitution, companion measures) are what students and checkers need, and they are generated from your numbers, not a canned example.
Winning move 3: the full companion set with a verdict. Chord, sector, segment, and sagitta in one result, plus a minor vs major arc label. A segment or sagitta usually means opening a second calculator on other sites.
One honest limit: this page covers circular arcs. The length of a general curve (a parabola, a sine wave) needs integral calculus, L = integral of sqrt(1 + (dy/dx)^2) dx, evaluated numerically for most functions. That is a different tool for a different problem.
Frequently asked questions
s = r x theta, where r is the radius and theta is the central angle in radians. If your angle is in degrees, convert it first: theta = degrees x pi / 180, or equivalently s = (theta_deg / 360) x 2 pi r. With r = 10 and theta = 90 degrees, s = 10 x 1.5708 = 15.708.
Multiply the radius by the angle’s share of the full circle. Worked example: radius 10, angle 90 degrees. The share is 90 / 360 = 0.25, the circumference is 2 pi x 10 = 62.8319, and 0.25 x 62.8319 = 15.708. The calculator does the same thing through the radian route and shows both steps.
The arc follows the curve; the chord is the straight line between the same two endpoints. The chord is always shorter: at radius 10 and 90 degrees, the arc is 15.708 and the chord is 14.1421. The gap grows with the angle, until at 180 degrees the chord is the diameter (20) while the arc is 31.4159.
A sector is the pie-slice region bounded by two radii and the arc between them. Its area is (1/2) r^2 theta with theta in radians: the same fraction of the full circle area (pi r^2) as the angle is of 2 pi. At r = 10 and 90 degrees the sector area is 78.5398, exactly a quarter of 314.1593.
The sagitta is the height of the arc above the midpoint of its chord: h = r – r cos(theta / 2). It is the number arch, camber, and lens problems ask for. At r = 10 and 90 degrees the sagitta is 2.9289, so the arc rises about 2.93 units above the straight line joining its ends.
Not for a single arc on one circle: a full turn is 360 degrees (2 pi radians) and the longest possible arc is the circumference itself. This calculator caps the angle there and, in angle-solving mode, rejects an arc length longer than the circumference. Problems about multiple rotations split into full turns plus a remainder arc.
The number follows the radius unit. Measure r in centimetres and the arc is in centimetres; measure in inches and the arc is in inches. Angles are unitless ratios either way, which is why degrees and radians give the same arc once converted correctly.
Divide the arc length by the angle in radians: r = s / theta. An arc of 15.708 cut by a 90 degree angle (1.5708 rad) gives r = 15.708 / 1.5708 = 10. Pick Radius in the Solve for menu and the calculator runs exactly this division.
Circumference is the special case where the arc is the whole circle: theta = 2 pi, so s = 2 pi r. Every other arc is a fraction of it, and the calculator reports that fraction as the Share of full circle line in the results.
Add This Calculator to Your Website
References
- Wolfram Research, “Arc Length, Wolfram MathWorld”. MathWorld: Arc Length. Accessed October 2026.
- Wolfram Research, “Circular Sector, Wolfram MathWorld”. MathWorld: Circular Sector. Accessed October 2026.
- Wolfram Research, “Circular Segment, Wolfram MathWorld”. MathWorld: Circular Segment. Accessed October 2026.
- Wolfram Research, “Semicircle, Wolfram MathWorld”. MathWorld: Semicircle. Accessed October 2026.
- National Institute of Standards and Technology, “Digital Library of Mathematical Functions, Chapter 4: Elementary Functions (inverse trigonometric and radian definitions)”. NIST DLMF: Chapter 4, Elementary Functions. Accessed October 2026.
- OpenStax, Rice University, “Calculus Volume 2: arc length and surface area sections”. OpenStax: Calculus Volume 2. Accessed October 2026.