Arc length in degrees:
formula and worked example

Distance along a curve, not across it

An arc is a portion of a circle's edge. Its length is the distance you would cover walking along the curve, as opposed to the straight line between its end points. Arcs appear in curved retaining walls, track bends, arched doorways, clock-hand tip travel, pipe bends, cut-and-fold paper cones and fan-shaped trim.

Because the arc belongs to a circle, you only need two things: how far the arc is from the centre, which is the radius, and how much of a full turn it spans, which is the central angle. Together they fix the length with no further information.

It is worth separating three related lengths that people mix up: the arc along the curve, the chord straight across, and the circumference all the way round. The arc sits between the chord and the circumference, and which of them applies depends on whether you are following the curve, spanning it or enclosing it.

The formula and its two inputs

Arc length equals 2 x pi x radius x angle / 360. Read it in two steps: 2 x pi x r is the whole circumference, and angle/360 is the fraction of that circumference you want. The angle must be the central angle between the two radii that end the arc.

If your angle is in radians the formula is even simpler: arc length equals radius times the angle, which is part of why mathematicians prefer that unit. In degrees, remember the 360 or the answer will be too large by a factor equal to the number of degrees.

A mental shortcut is to note that each degree of arc is about 0.01745 times the radius. Multiply that by the number of degrees and the radius and you have the answer to within a fraction of a per cent, which is enough for rough site work.

A 72 degree arc with radius 5

Imagine a curved edging strip that follows a circle of radius 5 units through 72 degrees.

1. 1Full circumference: 2 x 3.14159 x 5 = 31.416.
2. 2Fraction of the circle: 72 / 360 = 0.2, which is one fifth.
3. 3Multiply: 31.416 x 0.2 = 6.283.
4. 4State the unit: 6.283 units of length.

A pleasing check: 72 degrees is one fifth of a turn, and one fifth of 10 x pi is exactly 2 x pi, which is 6.283. Five such arcs would close up the full circle, which is how a regular pentagon is laid out.

Arc versus chord in real work

The straight-line chord joining the two ends of this arc is 2 x r x sin(angle/2) = 10 x sin 36 degrees, about 5.88 units. The arc itself is 6.28. The 7% gap is the extra material a curved piece needs over a straight cut between the same points, and it widens as the angle increases. For small angles (under about 10 degrees) the two are nearly equal, so a straight approximation is usually fine.

When you buy or cut flexible material for an arc, add allowance for fixing, joints and any waste at the ends. For a rigid curved element such as a bent pipe or rolled steel section, the relevant length is normally measured along the centre line, which means using the radius of that centre line rather than the inner or outer face.

Radius up by 10%

Moving from 5 to 5.5 in radius lifts the arc from 6.283 to 6.912, exactly 10% more. Angle behaves the same way: 144 degrees would double the 72 degree length. Unlike area formulas there is no squaring here, so planning is straightforward and errors do not compound. If an arc is too short by a given percentage, increase either input by that same percentage.

This also explains why outer lanes of a circular track must be staggered. Each extra unit of radius adds the same fraction of 2 x pi to every corner, so equalising distances needs a predictable offset.

Errors that catch people out

• Forgetting the division by 360, or using 180 by mistake.
• Using the chord length as the arc length. For a 72 degree arc this under-orders by about 7%.
• Using a radius measured to the wrong edge, for instance the outer face of a wall when the line you need lies along its centre.
• Mixing radians and degrees. 1.2566 radians is 72 degrees; entered as degrees it gives an arc of only about 0.11 units.

Scenario: edging a curved garden path

A path bends through a quarter turn (90 degrees) at a radius of 3 m measured to its centre line. The centre-line length is 2 x pi x 3 x 90 / 360 = 4.71 m. If the path is 1 m wide, the inner edge has a 2.5 m radius and a length of 3.93 m, while the outer edge at 3.5 m radius is 5.50 m long. Ordering one length for all three would leave the outer edging short by nearly 0.8 m, which is the practical reason to do the arc calculation edge by edge.

The same principle applies to a bent pipe or rolled bar. The length to cut is the arc along the neutral line, not the outside edge, so a bend of 90 degrees at a centre-line radius of 300 mm needs about 471 mm of straight stock, before any allowance for the straight tangents.