Arc Length Calculator

What the arc is, and what the formula says

Arc length is the distance along a curve, and for a circular arc the curve is part of a circle with a radius r and a central angle theta. The arc length s is the length of the curved slice itself, not the straight line across it and not the area of the wedge.

The formula is one line: s = r theta, with theta in radians. The OpenStax Precalculus text on angles explains why that works, because a radian is defined as the angle whose arc length equals the radius. One radian on a circle of radius 10 has an arc of exactly 10 units, and two radians have an arc of 20.

That is the idea behind the calculator here: convert the angle to radians, multiply by the radius, and read off the arc.

One reason the radian version matters is that it stays true when the radius changes. A 60 degree arc is one sixth of any circle, so it is always 2 pi r / 6, whether r is 3 or 3000.

Degrees, radians and the two ways to write it

Most people carry angles in degrees, and every tool that matters expects radians, which is where the classic mistake comes from. Feeding 90 into s = r theta without converting gives an arc 57 times too long, because 90 radians is 5156.62 degrees, more than 14 full turns.

The conversion is a divide and a multiply by pi: theta in radians = angle in degrees / 180 x pi. The arc length notes from the University of Nebraska work through the same derivation and reach the same one line formula. The table below puts the same radius of 10 against the angles people use most.

The last row is the full circumference, 2 pi r, and it makes a good check on any arc figure. If your arc is longer than the circumference, then the angle or the radius is wrong.

Some fields measure angle in gradians, where a full turn is 400 units, and others in turns, where a full turn is 1. This calculator takes degrees and radians, which covers nearly every school and trade calculation.

Worked example: 90 degrees on a 10 cm radius

Take a wheel rim with a radius of 10 cm and an angle of 90 degrees. Step one is the conversion: 90 / 180 x pi is 1.5708 radians. Step two is the formula: s = r theta, so 10 x 1.5708 is 15.71 cm. That is the length of the curved quarter circle, and it is longer than the straight chord of 14.14 cm that joins the same two points.

The chord comes from 2 r sin(theta / 2): 2 x 10 x sin(0.7854) is 14.14 cm. The sector area comes from 0.5 r squared times theta: 0.5 x 100 x 1.5708 is 78.54 square cm. The full circle is 2 pi r, which is 62.83 cm around, and the 90 degree arc takes exactly a quarter of it, so 62.83 / 4 is 15.71 cm again.

That last check is the useful one for the classroom, because a quarter of the circumference has to be a quarter of the total. It is also the check to use when someone quotes an arc length that does not sit inside its own circle.

Chord, sector and segment

The arc is one of four things you can work out from a radius and an angle, and mixing them up is the usual source of wrong answers.

The segment is the crescent shape between the arc and the chord, and it is the area many people picture when they think of a slice of pie. At a 90 degree angle on a radius of 10 it is 78.54 minus 50, which is 28.54 square cm.

The gap between arc and chord grows with the angle. At 30 degrees the arc is 5.24 and the chord is 5.18, a difference of 0.06. At 180 degrees the arc is 31.42 and the chord is 20, a difference of 11.42. The two numbers sit close together for small angles and far apart for large ones, which is why the small angle approximation fails for half circles.

Keep the units straight as well. The area formulas use squared length units, so a radius in centimetres gives an area in square centimetres while the arc stays in centimetres.

Working backwards

The arc length formula rearranges as easily as it applies. Given the arc and the radius, the angle in radians is theta = s / r, and multiplying by 180 / pi gives degrees. An arc of 15.708 units on a radius of 10 is 1.5708 radians, which is 90 degrees. Given the arc and the angle, the radius is r = s / theta, so an arc of 12 units at 2 radians sits on a radius of 6.

Both reverse modes are in the calculator above. Worth using when a supplier quotes a belt length or a pipe bend, because a drawing can muddle the arc and the chord, and the difference grows fast with the angle. At 90 degrees the gap is 1.57 units on a radius of 10. At 170 degrees the arc is 29.67 and the chord is 19.92, a gap of 9.75 units on the same radius.

There is a related question the formula cannot answer: the radius of a circle that fits a given arc and a given chord. That one needs the chord formula and a little algebra, and it is the shape of the problem that comes up when you are bending a pipe or planning a garden path.

Real arcs: Earth, wheels and pizza rims

One degree of latitude on the Earth is a good check on the arithmetic. The NASA NSSDCA Earth fact sheet lists a volumetric mean radius of 6371.0 km, and one degree of arc on that sphere is 6371 x pi / 180, which is 111.19 km. A quarter turn along a meridian is 10,007.54 km, and the full equator is 40,030.17 km.

A degree of longitude shrinks as you move away from the equator; a degree of latitude does not.

Everyday arcs are everywhere. A pizza with a 12 inch diameter has a 6 inch radius, so the half rim from one side to the other is 180 degrees and 6 x pi, which is 18.85 inches of crust edge. A bicycle wheel of 26 inch diameter has a 13 inch radius, and one turn of the rim is 2 pi x 13, which is 81.68 inches of rubber passing through the contact patch.

Belts and pulleys are the classic trade application. A 3 inch pulley turning at 200 revolutions per minute moves its belt 3 pi x 200, which is 1884.96 inches, or 157.08 feet, every minute. Double the diameter and the belt moves twice as far for the same number of turns, which is why step pulleys change speed and cost you belt length at the same time.

Arcs on roads, rails and pipes

Engineering uses arc length for circular curves, and there the units and the precision both matter. A road curve with a 500 foot radius that turns 30 degrees has an arc of 500 x pi / 6, which is 261.80 feet, and the chord between the same two points is 2 x 500 x sin(15 degrees), which is 258.82 feet. The tangent from the point of curvature to the point of tangency is 500 x tan(15 degrees), which is 133.97 feet. Designers work with the curve length because that is what the traveler drives, and with the tangent because that is what the layout crew stakes in the ground.

Railway and highway work measures the curve along the arc rather than along the chord, and some old tables use the chord instead, so check which convention a table follows before you mix numbers from two sources.

Pipe bending works the same way. The pipe you need for a bend is the arc plus an allowance for the fittings on each end, and a long radius bend takes more pipe than a short radius bend for the same turn. Being out by a few degrees on a large radius is a lot of pipe.

Common mistakes and edge cases

The list of things that go wrong is short.

Edge cases: an angle of zero gives an arc of zero. An angle of 360 degrees gives the full circumference, 62.83 units at a radius of 10. An angle above 360 degrees describes more than one turn of the same circle, and the calculator stops there.

The sources behind this page are the OpenStax Precalculus chapter on angles, the University of Nebraska arc length notes, and the NASA NSSDCA Earth fact sheet. Everything on this page runs in your browser and nothing you type is uploaded or stored.

One last trap: distance along a sphere is not distance on a flat page. Over a quarter of the Earth's circumference, the arc along the surface and the straight line between the same points differ by a lot, and that straight line runs deep through the planet.