Arc Length Calculator — the curved distance along a circle

The arc length calculator measures one thing precisely: the distance you would travel if you walked along the curved edge of a circle, from one end of an angle to the other. It is not the straight-line distance across the opening, and it is not the area of the wedge underneath. It is the curve itself, and it is the number you need whenever a physical thing has to follow that curve — a strip of edging, a length of band, a cut of trim, a run of track.

Arb Digital publishes this alongside the rest of our free geometry set because arc length is one of the few school formulas that people meet again in real work and then get wrong under pressure. The mistake is almost always a unit mistake, not a maths mistake. This calculator takes degrees, radians, gradians or turns, converts internally, and prints the working with your own numbers so you can see which conversion was applied.

What This Arc Length Calculator Does

Give it a radius and a central angle and it returns the arc length as the headline figure. Underneath it shows the chord length — the straight line joining the two ends of the arc — the area of the sector that the arc encloses, the angle restated in radians, and the circumference of the complete circle so you can see what fraction of it your arc represents.

A radius of 8 cm with a central angle of 120° gives an arc length of about 16.7552 cm, a chord of 13.8564 cm, a sector area of 67.0206 cm², an angle of 2.0944 radians, and a full circumference of 50.2655 cm. The arc is exactly a third of that circumference, which is the sanity check: 120° is a third of 360°, so the arc must be a third of the way round.

The mode selector at the top does something most arc calculators skip. You can enter a measured arc and a known radius to recover the angle, or a measured arc and a known angle to recover the radius. That is the direction the problem actually runs in when you are working from something physical — you have a curved edge you can put a tape along, and you want to know what circle it came from.

How to Use It

1. Choose what you know. The default solves for the arc from a radius and an angle. The other two modes work backwards from a measured arc.
2. Enter the radius from the centre of the circle to the curve. If your drawing gives a diameter, halve it before entering — this is the single most common source of a doubled answer.
3. Enter the central angle and pick its unit. Degrees and radians are both first-class here; gradians and turns are there for surveying and rotational work.
4. Select a length unit. Every length in the results uses it, and the sector area uses the same unit squared.
5. Read the working line underneath. It restates the conversion and both forms of the formula with your figures, so you can check the result rather than trust it.

The Formula: How Arc Length Is Calculated

In radians the formula is as short as formulas get: s = rθ. Arc length is the radius multiplied by the angle. With r = 8 cm and θ = 120° = 2π/3 ≈ 2.094395 radians, s = 8 × 2.094395 = 16.7552 cm.

In degrees the same quantity carries a conversion factor: s = 2πr × (θ / 360). That reads more plainly — take the whole circumference and keep the fraction of it your angle covers. The same example gives 2π × 8 × (120/360) = 50.2655 × 0.3333 = 16.7552 cm. Identical, as it must be, because the two expressions are the same statement written in different units.

The chord that closes the arc is c = 2r sin(θ/2) = 2 × 8 × sin 60° = 16 × 0.866025 = 13.8564 cm. The sector area beneath the arc is A = ½r²θ = ½ × 64 × 2.094395 = 67.0206 cm². MathWorld's arc entry sets out these relations formally, along with the classical theorems that connect chords and the arcs they cut off.

Degrees and Radians: Why One Formula Has No Constant

A radian is defined by the arc itself. It is the angle you get when the arc length equals the radius — wrap the radius around the circumference and the angle it subtends is one radian, about 57.2958°. Because the definition is built from arc length, the arc length formula in radians has nothing left to convert: s = rθ, no π, no 360.

Degrees are a human convention. Three hundred and sixty is a useful number because it divides cleanly by two, three, four, five, six, eight, nine, ten, twelve and more, which makes common fractions of a circle land on whole degrees. It carries no geometric meaning, so every circular formula written in degrees has to buy the conversion back in the form of a π/180 or a /360 somewhere.

The practical rule for anything beyond this page: convert to radians first, do the geometry, convert back at the end if your reader expects degrees. This calculator does exactly that internally. If you need the conversion by itself for other work, our angle converter handles degrees, radians, gradians, turns and minutes of arc. MathWorld's radian entry covers why the unit behaves this way in calculus as well as geometry.

One trap deserves naming. Most programming languages and spreadsheet functions take radians, not degrees. A trigonometric function fed a raw degree figure will return a number rather than an error, and that number will be silently wrong. If a calculation involving angles is producing plausible but incorrect output, the conversion is where to look first.

Arc, Chord and the Gap Between Them

The arc is always longer than the chord, because a curve between two points is longer than the straight line joining them. What is worth knowing is how the gap behaves, because it decides when the difference matters and when you can ignore it.

At small angles the two are nearly identical. A 10° arc on a radius of 8 is 1.3963 long, against a chord of 1.3945 — a difference of about 0.13%. At 60° the arc is 8.3776 and the chord is 8, a gap of 4.7%. At 120° the arc is 16.7552 and the chord 13.8564, a gap of 20.9%. At 180° the arc is a half-circumference of 25.1327 against a chord of just 16, the diameter — a gap of 57%.

That is why polygon approximations of circles work well with many short sides and badly with few long ones, and it is why a curved wall measured as a series of straight tape pulls always comes up short of the true developed length. The error compounds in the direction of ordering too little material. If you are estimating a curved run and can only measure straight lines, measure in more, shorter steps than feels necessary. MathWorld's chord entry covers the related geometry, including the sagitta — the height of the arc above the chord, which is often the easiest thing to measure on site.

Working Backwards From a Measured Arc

Real problems frequently run the other way. You have a curved edge in front of you and you need the circle behind it, either to reproduce the curve or to check that it is a true arc at all.

If you know the radius and can measure the arc, the angle is θ = s / r in radians, or (s / r) × 180/π in degrees. If instead you know the angle — a quadrant is 90°, a segment of a six-part division is 60° — the radius is r = s / θ, again with θ in radians. Both are available in the mode selector above, and the working line shows the rearrangement it performed.

This is the calculation behind laying out a curve with a string and a peg. Measuring an arc of 16.755 units and knowing it should span a third of a circle gives a radius of 16.755 ÷ 2.0944 = 8 units, which is the string length you set. It is also the check that tells you a curve is not circular: measure two different portions of it that should span equal angles, and if they return different radii the curve is an ellipse, a spline, or a mistake. For the elliptical case, arc length has no closed-form solution at all and must be computed numerically, which is why our ellipse calculator uses an approximation for perimeter rather than an exact formula.

Where Arc Length Actually Gets Used

Curved trim and edging is the everyday one. A bay window, a curved worktop nosing, a round patio border or the fascia on a curved bar all need material cut to the developed length of the curve, not to the straight span. Order to the chord and you are short by the gap described above.

Cone and duct development is the one where arc length earns its keep. Unroll a cone and it flattens into a sector: the sector radius is the cone's slant height and the sector's arc length equals the circumference of the cone's base. That single equality is what lets a sheet-metal pattern be set out flat and then rolled. The same logic gives the flat pattern for a lampshade, a funnel, or a fabric gore.

Wheels and belts turn the formula into distance travelled. A wheel of radius r turning through θ radians without slipping advances exactly rθ along the ground, which is why odometers are calibrated to a rolling radius and why fitting larger tyres makes a speedometer read low. Belt and pulley lengths use the arcs of contact on each pulley plus the straight spans between them.

Running tracks are a well-known application of the same idea. Each lane's bends are arcs of a larger radius than the one inside it, so a full lap in an outer lane is longer even though the straights are identical. That difference is exactly why staggered starts exist. Curved staircases, radial paving, road bend layout, curved conveyor sections and the sweep of a windscreen wiper all resolve to the same rθ.

Arc, Sector or Whole Circle: Which Page You Need

Three of our tools cover circles and they deliberately do not overlap: this page gives the length of the curved edge, our sector area calculator gives the area of the wedge that edge encloses along with the segment cut off by the chord, and the circle calculator handles the complete circle's radius, diameter, circumference and area.

In practice the question to ask is what you are actually buying or building. If it is a length of something — edging, band, cable, trim — you want an arc. If it is a surface — turf, paint, tiling, a fabric panel — you want a sector or a segment. If the angle is a full 360° there is no arc to speak of and you want the whole circle.

For a mixture of straight and curved edges, our perimeter calculator covers the standard straight-sided shapes, and the polygon calculator handles any number of sides. Add the arc from this page to the straight runs from those to get a total boundary for a composite outline such as a stadium shape or a rounded rectangle.

Common Mistakes to Avoid

• Putting degrees into s = rθ — entering 120 instead of 2.0944 overstates the arc by a factor of about 57. The formula only holds in radians.
• Entering the diameter as the radius — arc length is directly proportional to the radius, so this doubles every length on the page.
• Cutting material to the chord — the straight span across the opening is always shorter than the curve, by 21% at 120° and more beyond that.
• Confusing the arc with the sector perimeter — the perimeter of the wedge is the arc plus the two radii, which is a much larger number.
• Reusing a radius from a different reference point — an inside radius and an outside radius on the same curve differ by the material thickness, and so do their arcs.

Related Free Tools From Arb Digital

Get the wedge area behind the curve with the sector area calculator, solve the whole shape with the circle calculator, switch angle units with the angle converter, convert a finished length with the length converter, and convert a finished area with the area converter. The complete set is listed in our free online tools hub.

Frequently Asked Questions