The Arc Length Formula, Explained with Examples

Arc length is the distance you would travel if you walked along a curved slice of a circle, rather than straight across it. Once you know the radius of the circle and the central angle that opens up the arc, you can find that curved distance with one short formula.

What is arc length?

An arc is a portion of the edge of a circle, like a single bite taken out of the rim of a wheel. Arc length is simply how long that curved piece is if you could straighten it out into a line. It is always measured along the curve itself, not as a straight shortcut between the two endpoints.

The size of an arc depends on two things: how big the circle is (the radius) and how wide the central angle is. The central angle is the angle at the center of the circle, between the two radius lines that reach the ends of the arc. A wider angle sweeps out a longer arc, and a bigger radius stretches that same angle over a longer distance.

The arc length formula

There are two common forms of the same formula. Which one you use depends only on whether your angle is measured in radians or in degrees.

Both versions describe the same idea: an arc is a fraction of the way around the circle. A full circle is 360 degrees and its edge measures the circumference, which is 2 x pi x r. So the degree form just takes the slice theta / 360 of that full circumference. The radian form is even tighter, because in radians the angle already encodes that fraction directly, so you only multiply by the radius.

Why the radian version is so simple

A radian is defined so that an angle of one radian sweeps out an arc exactly equal to the radius. That definition is what makes s = r x theta work with no extra constants. A full circle is 2 x pi radians, and 2 x pi x r is the circumference, so everything lines up. This is also why higher math almost always prefers radians over degrees. If you want a refresher on where the circumference comes from, see the circumference formula.

How to find arc length step by step

Suppose a circle has a radius of 5 units and a central angle of 60 degrees. Here is how to find the arc length using the degree form, s = 2 x pi x r x (theta / 360).

1. Start with the formula: s = 2 x pi x r x (theta / 360).
2. Plug in the values: s = 2 x pi x 5 x (60 / 360).
3. Simplify the fraction: 60 / 360 equals 1/6, so the arc is one sixth of the full circle.
4. Find the full circumference: 2 x pi x 5 is about 31.42.
5. Take one sixth of it: 31.42 divided by 6 is about 5.24.
6. So the arc length is about 5.24 units.

You can check this with the radian form. First convert 60 degrees to radians by multiplying by pi / 180, which gives about 1.047 radians. Then s = r x theta is 5 x 1.047, which is also about 5.24. Same arc, same answer, just a different angle unit. Getting the same number two different ways is the best confirmation that you have not slipped up on units, which is where almost every arc length mistake hides.

A second worked example in radians

Now try one where the angle is already in radians, so you can use the shortest form directly. Suppose a circle has a radius of 9 centimeters and a central angle of 2 radians. Because the angle is in radians, reach straight for s = r x theta.

1. Start with the radian formula: s = r x theta.
2. Plug in the values: s = 9 x 2.
3. Multiply: 9 x 2 equals 18.
4. So the arc length is 18 centimeters, with no pi and no conversion needed.
5. Sanity check: 2 radians is a little under a third of the full 2 x pi radians, so this arc should be a little under a third of the circumference, 2 x pi x 9 which is about 56.5. One third of that is about 18.8, and 18 is just under it, so the answer is reasonable.

Notice how much faster the radian route is when the angle arrives in radians. There is no fraction, no 360, and no circumference to compute first. This is exactly why physics, calculus, and engineering default to radians: the arithmetic is cleaner and the formula carries less baggage.

Converting between degrees and radians

Because the two formulas use different angle units, knowing how to convert is essential. The whole conversion rests on one fact: a half circle, 180 degrees, equals pi radians.

• Degrees to radians: multiply the degree value by pi / 180.
• Radians to degrees: multiply the radian value by 180 / pi.
• Quick checks: 90 degrees is pi / 2 radians, 180 degrees is pi radians, and 360 degrees is 2 x pi radians.

If your angle is already in radians, reach straight for s = r x theta because it skips a step. If it is in degrees, you can either convert first or just use the degree form directly. Both routes give identical results, so pick whichever keeps your work cleaner.

Worked examples at a glance

The chart below shows arc lengths for a few common angles, all calculated with the degree form and pi as about 3.14159. Notice how the arc length is always the same fraction of the circumference as the angle is of 360 degrees.

The last row is a useful sanity check: a 360 degree arc is the entire circle, so its arc length should equal the full circumference, 2 x pi x 8, which is about 50.27. Whenever your angle hits 360 degrees, the arc length and the circumference are the same number.

Working backward from arc length

You can rearrange the formula to solve for whatever piece is missing. If you already know the arc length, a little algebra recovers the angle or the radius.

• To find the angle in radians: theta = s divided by r.
• To find the radius: r = s divided by theta (with theta in radians).
• To find the angle in degrees: theta = (s divided by the circumference) times 360.

For example, if an arc is 5.24 units long on a circle of radius 5, then theta = 5.24 divided by 5, which is about 1.047 radians, or roughly 60 degrees. This reverse trick is handy in design and engineering, where you often know the curved distance you need and have to back out the angle or the size of the circle.

A practical case: imagine you are laying out a curved garden path that must be exactly 12 meters long, and the curve has to fit a flower bed of radius 8 meters. Solving theta = s divided by r gives 12 divided by 8, which is 1.5 radians, or about 86 degrees. Now you know precisely how wide to open the curve on the ground. The same back-calculation tells a machinist how far to rotate a part, or a track designer how to mark a lane so each runner covers the right distance.

Common mistakes to avoid

• Mixing up degrees and radians: s = r x theta only works when theta is in radians. Putting 60 into it instead of 1.047 gives a wildly wrong answer.
• Forgetting to convert: if your angle is in degrees, either convert with pi / 180 or use the full degree form, not the bare radian formula.
• Confusing arc length with chord length: arc length follows the curve, while a chord is the straight line between the endpoints. The chord is always shorter.
• Rounding pi too early: keep extra decimals during the steps and round only the final answer so small errors do not pile up.
• Mismatched units: if the radius is in cm, the arc length is in cm, not square cm. Arc length is a distance, not an area.

Practical tips for getting it right

Beyond avoiding the common errors, a few habits make arc length problems faster and more reliable, especially under exam pressure or when the numbers are messy.

• Read the angle units first. Before touching a formula, check whether theta is in degrees or radians, then choose the matching form. This one habit prevents the most common wrong answer.
• Keep pi symbolic when you can. Writing the answer as a multiple of pi, such as 5 x pi / 3 for the 60 degree example, is exact and avoids rounding drift until the very end.
• Estimate before you compute. A 90 degree arc is a quarter of the circumference, and a 180 degree arc is half. A rough mental estimate flags a typo before it costs you marks.
• Match the radius unit to the answer. Centimeters in means centimeters out, so label the result so you do not lose track of the unit.
• Use the same value of pi throughout. Mixing 3.14 in one step and 3.14159 in another introduces small inconsistencies that can matter in engineering work.

Good to know: arc length, sectors, and real uses

Arc length is closely tied to the sector of a circle, which is the pie-slice region bounded by two radii and the arc. The arc is the curved crust of that slice. While arc length measures only the curved edge, the sector area measures the filled-in region, so do not confuse the two even though they share the same angle.

Arc length shows up in more places than you might expect. It tells you how far a point on a spinning wheel travels, how long a curved running track lane is, how much trim a fan-shaped window needs, and how far a robot arm sweeps through an angle. The same logic that powers a steady velocity formula on a straight road extends to circular motion, where distance traveled around a curve is exactly an arc length.

• Wheels and gears: a point on the rim travels one arc length per slice of rotation.
• Tracks and curved paths: the inside and outside lanes have different radii, so they have different arc lengths for the same angle.
• Pizza and pie slices: the curved crust of a slice is an arc length.
• Architecture and design: curved arches, windows, and trim are measured as arcs.
• Navigation and maps: distances along a curved route or the edge of a roundabout are arc lengths.

If you find yourself working with the straight-line distance between the two endpoints instead of the curved edge, that is a chord, not an arc, and it follows different math. When a shape mixes straight and curved segments, such as a curved track with straight finish lines, you may also need the perimeter formula to add every side together correctly.

Calculate it instantly

Want the answer without doing the arithmetic by hand? Use the circle calculator below to work with radius, circumference, and circle measurements, then apply the angle fraction to get your arc length in seconds.

Once arc length clicks, the rest of circle geometry falls into place quickly, since it all shares the same radius. The area of a circle is the natural next step, and the circumference is just the special case where the angle is a full 360 degrees. Master those three and you can handle almost any circle problem with confidence.