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CalcMax

Arc Length Calculator

Range: 0 cm – 1,000,000,000 cm

Range: 0 ° – 360 °

Result

7.8540 cm

Arc length

An arc length calculator takes the radius of a circle and an angle at its centre and returns how long the piece of the circle is, in centimetres. A circular arc is the curved part of the circle between two points on it, and the angle that names it is the one at the centre between the two radii drawn to those points. The arithmetic is a proportion and nothing more: the whole circle is 360 degrees and its whole outline is 2πr long, so an arc of θ degrees is θ/360 of that. Whatever fraction of the circle the angle covers, the arc covers the same fraction of the circumference. The reason this is worth its own page rather than being left to a full circle calculator is that most of the places a curve turns up are not whole circles. A bend in a road is an arc; so is the path of a door handle, the sweep of a windscreen wiper, a curved counter, an arch, the stretched tape on a dial. In every one of those the question is the length of the curved part, because that is what gets cut, priced, paved or driven. There is one trap in this subject and it is a unit trap. The compact form of this formula is L = rθ, and in that form θ is in radians, not degrees — radians being the unit that exists precisely to make that line come out without a factor in front of it. Feed 60 degrees into L = rθ and the answer comes back about 57 times too large. It will not look wrong, either: it will look like a long arc. This page takes degrees, because 60 is easier to type than 1.0472, and the formula it computes is written with the 360 in it so that the degrees are accounted for before the multiplication happens.

The length of an arc from its radius and the angle at the centre

Radius (cm)Central angle (degrees)Arc length (cm)
5605.236
5907.854
518015.708
536031.4159
109015.708
612012.5664
1227056.5487
500

Eight arcs and three columns. The first row is the pair the page loads with, a sixth of a circle. The fourth is the row that checks the page against the rest of the site: at 360 degrees the arc has closed into a full circle, so it has to equal the circumference, and 31.4159 is what the circle and circumference pages print for a radius of 5. The fifth row is the one worth pausing on — a radius of 10 at 90 degrees gives the same 15.708 as a radius of 5 at 180 degrees, because doubling one and halving the other leaves the product alone. The sixth is the only row whose answer is an exact multiple of π: a radius of 6 at 120 degrees is 4π, which is 12.5664. Every value here is recomputed from its two inputs when the page is built, and because a sector page computes the same quantity from the same two numbers, the arc-length column of this table and of that one should read identically line for line.

Formula

L = (θ ÷ 360) × 2πr

Radius
The distance from the centre of the circle to the arc, in centimetres
θ (theta)
The angle at the centre, in degrees — the angle between the two radii that meet the ends of the arc. Not the angle the arc itself turns through on the page, and not the angle between the two ends as seen from outside
360
The whole circle in degrees, and the reason this formula is written the way it is. Dividing the angle by 360 turns it into the fraction of the circle the arc covers
2πr
The circumference — the length of the entire circle, which is what is being taken a fraction of. The π is the same constant every circle has, and r is the radius again
L
The length of the arc, in centimetres. The same fraction of the circumference as the angle is of 360 degrees
L = rθ
The same formula in its compact form, which is the one textbooks and reference works print. In this form θ is in radians, not degrees — using 60 here instead of 1.0472 is the mistake the whole page warns about
Four decimal places
How wide the reading is written. The answer usually cannot be written down exactly, because π cannot — a quarter of a circle of radius 5 comes to 7.8540 and stops there

Anything that is part of a circle and whose length matters. The plainest case is measurement: a curved wall, a curved counter, a bay window, a bend in a path, the length of a fence that follows a curve rather than a line — the length along the curve is what gets ordered, and it is longer than the straight distance between the two ends. Materials that come in strips and get bent are the second case: edging, trim, weatherstrip, conduit, the curved section of a handrail, a piece of sheet metal rolled to a radius. Working out a length before cutting is the entire point, and an arc cut short is scrap. Vehicles and machines are the third: the distance a wiper sweeps, the path of a robotic arm, a belt running over a curved guide, the length of a train of chain around a sprocket. In each of those the arc length is the distance actually travelled, and it is what the motor has to be sized for. Then there is the classroom use. The proportion is easy to see on a circle drawn with a compass — a 90-degree arc is a quarter of the circumference, and you can check it against the formula — and it is the piece of machinery that lets you go on to sectors, segments and eventually to radians, which are nothing more complicated than the same proportion stated the other way round.

Worked examples

  1. A radius of 5 and an angle of 90 degrees

    1. The angle as a fraction of the circle: 90 ÷ 360 = 1/4
    2. The whole circumference: 2 × π × 5 = 31.4159
    3. A quarter of it: 31.4159 ÷ 4 = 7.854

    A quarter of a circle, and the pair the page loads with. The two readings to keep in mind together are this 7.854 and the 31.4159 the same radius gives at 360 degrees — one is exactly four times the other, which is a quick way to see that the angle really is being used as a fraction and not as something else.

  2. A radius of 5 and an angle of 60 degrees

    1. The angle as a fraction of the circle: 60 ÷ 360 = 1/6
    2. The whole circumference: 2 × π × 5 = 31.4159
    3. A sixth of it: 31.4159 ÷ 6 = 5.236

    A sixth of a circle, which is the angle an equilateral triangle makes at the centre and the reason 60 degrees turns up whenever something is divided into six. This is the row the sector page also prints, so the two pages can be put side by side: whatever the arc length comes to here, the sector page shows the same figure for the same radius and angle.

  3. A radius of 5 and a whole turn

    1. The angle as a fraction of the circle: 360 ÷ 360 = 1
    2. The whole circumference: 2 × π × 5 = 31.4159
    3. All of it: 31.4159

    The boundary case, and a check on the whole page: at 360 degrees the arc has gone all the way round and closed up, so it is the circumference, and the circumference of a circle of radius 5 is 31.4159. If this row ever printed anything else, the formula would be wrong — that is what makes it worth putting on the table.

  4. A radius of 10 and an angle of 90 degrees

    1. The angle as a fraction of the circle: 90 ÷ 360 = 1/4
    2. The whole circumference: 2 × π × 10 = 62.8319
    3. A quarter of it: 62.8319 ÷ 4 = 15.708

    The same quarter-circle as the first example and the same length as the 180-degree arc of radius 5. Doubling the radius and halving the angle leaves the arc unchanged, because the arc length depends on the two together rather than on either one. It is the cheapest demonstration there is of why the formula multiplies rather than adds.

  5. A radius of 5 and an angle of 0

    1. The angle as a fraction of the circle: 0 ÷ 360 = 0
    2. The whole circumference: 2 × π × 5 = 31.4159
    3. None of it: 0

    An angle of zero means the two radii have closed onto each other and the arc has shrunk to a single point on the circle, so its length is zero. That is a real answer and the page reports it, which is different from leaving a box empty — an empty box has nothing to work from and shows no result at all.

Limitations

This page returns the length of the arc and nothing else. It does not give the area of the sector, the length of the straight chord between the two ends, the angle, or the radius — the sector page covers the first two, and going backwards from an arc length to the angle is a different page. The angle is in degrees and is capped at 360: a central angle larger than a full turn is refused rather than wrapped round, because a central angle is by definition not more than one revolution. Zero is accepted for either input and gives an arc of length zero, which is correct for a point and is not the same as an empty box. Both inputs are taken in the same units, so a radius in inches gives an arc length in inches; nothing is converted. The answer is the length along a circular arc and says nothing about any other curve — an elliptical arch, a spiral, a catenary or a bend that is not a true circle will not come out right, and of those the ellipse is the case people expect to be easier than it is.

Frequently asked questions

Is the angle in degrees or radians?
Degrees. The formula this page computes is (θ ÷ 360) × 2πr, which is written so that the degrees are divided out before anything else happens. The compact form you may have seen elsewhere, L = rθ, takes radians instead, and that is worth knowing because entering 60 into it gives an answer about 57 times too large — and an answer that is merely long does not look wrong.
Which angle in the diagram is the central one?
The one at the centre of the circle, between the two radii drawn out to the two ends of the arc. It is not the angle between the two ends measured from somewhere on the arc, and it is not the angle the curve appears to turn through on the page. If your diagram has several angles marked, the central one is the one whose vertex is the centre.
How is this different from the sector calculator?
The sector page gives three readings for the same two inputs — the area of the wedge, the arc length and the straight chord between the ends — because somebody working on a sector usually wants more than one of them. This page gives the arc length alone. The two agree exactly on that one number, so if you only need the length, this is the shorter road.
Can the angle be more than 360 degrees?
No, and the page refuses it rather than wrapping it round. A central angle is by definition an angle at the centre of a circle, and it cannot exceed one full turn; 400 degrees would mean an arc longer than the whole circumference, which is not an arc. The limit is deliberate and matches the sector page, so the same pair of numbers is accepted or refused in the same way on both.
Does the arc length depend on the radius?
Yes, and the dependence is a multiplication rather than an addition. Doubling the radius doubles the arc, and doubling the angle doubles it too, so a 180-degree arc of radius 5 and a 90-degree arc of radius 10 are the same length. That is why the formula multiplies the two rather than doing anything else with them.
What if the curve is not a circle?
Then this page will not give you the right answer, and it will not warn you either. Every arc here is a piece of a true circle. An elliptical arch, a spiral staircase, a catenary curve and the path of a thrown object all have lengths that need different mathematics, and of those the ellipse is the one people most often assume is a simple fraction of a circle when it is not.

References

Related calculators

Central Angle CalculatorThe angle an arc subtends at the centre of its circle.Sector Area CalculatorThe slice of a circle cut by two radii: area, arc length and chord from the angle.Circumference CalculatorCircumference of a circle from its radius or diameter.Circle CalculatorRadius, diameter, circumference and area from any one of them.