Arc Length Calculator

Definition

The arc length s spans the central angle θ (in radians):

s = r \, θ

Converting degrees to radians

A full circle is 2π rad (360°):

θ_\mathrmrad = θ_\mathrmdeg · (π)/(180)

Chord length

The straight-line distance between the arc's endpoints (the "chord") is twice the radius times the sine of half the central angle:

c = 2r \, \sin\!(\tfracθ2)

Sector area

A "sector" is the pie-slice outlined by the arc plus two radii. Its area is the radius squared times half the central angle:

A_\mathrmsector = \tfrac12 r^2 θ

Why radians are the natural unit

The arc length formula s = rθ is cleanest when θ is in radians — no extra π/180 factor. The radian is also dimensionless (defined as arc length divided by radius), which keeps dimensional analysis consistent.

θ_\mathrmrad \equiv (s)/(r)

Working in Degrees

If the angle is given in degrees, convert first or fold the conversion into the formula. Since a full turn is 360° and its arc is the whole circumference 2π r, an arc of θ degrees is that same fraction of the circumference.

A worked case: a circle of radius 5 m subtending 72° gives s = (72/360)× 2π× 5 = 0.2 × 31.416 = 6.283 m. The same answer follows from converting 72° to 0.4π = 1.2566 rad and multiplying by 5. Checking the fraction of a full circle first is a good sanity test — 72° is a fifth of a turn, so the arc should be a fifth of the circumference.

s = \fracθ_\textdeg360 × 2π r = \fracπ r θ_\textdeg180

Arc Length, Sector Area and Chord

Three quantities describe the same slice of a circle and are easily confused. The arc is the curved edge, length rθ. The sector is the pie-slice region, area \tfrac12r^2θ. The chord is the straight line joining the arc's endpoints, length 2r\sin(θ/2).

Note the neat relationship between the first two: sector area is half the arc length times the radius, exactly parallel to a triangle's area being half its base times its height. The chord is always shorter than the arc, and the ratio approaches 1 as the angle shrinks — which is the geometric statement behind the small-angle approximation \sinθ ≈ θ. For a 10° angle the chord is already within 0.13% of the arc.

A_\textsector = \tfrac12r^2θ, \qquad c = 2r\sin\!((θ)/(2))

General Curves and Practical Uses

The circular formula is a special case of the general arc length integral, which measures the length of any smooth curve by summing infinitesimal hypotenuses. For a circle, that integral collapses to rθ; for most other curves it has no closed form and must be evaluated numerically — the ellipse being the classic example, whose perimeter requires elliptic integrals and is the reason approximations like Ramanujan's are still used.

The circular case appears constantly in practice. Highway and railway curves are laid out by arc length along a specified radius. A belt or chain wrap around a pulley uses s = rθ to find contact length, which in turn sets the friction available. Machining a curved slot, calculating the material needed to roll a cylindrical tank, and converting an angular encoder reading into linear travel are all the same computation.

L = ∫_a^b √(1 + (\fracdy)dx)^2\,dx

Worked Example

Inputs

• radius: 5
• angleDeg: 60

Result: arcLength: 5.236 chordLength: 5 sectorArea: 13.09 angleRad: 1.0472

Explanation

For r = 5 and θ = 60° (which equals π/3 ≈ 1.0472 rad): arc length s = 5 · π/3 ≈ 5.236, chord = 2 · 5 · sin(30°) = 5, sector area = ½ · 25 · π/3 ≈ 13.09.

Second Scenario

Inputs

• radius: 7.25
• angleDeg: 60

Result: arcLength: 5.236 chordLength: 5 sectorArea: 13.09 angleRad: 1.0472

Explanation

This scenario uses different inputs (radius = 7.25, angleDeg = 60) to show how changing one variable affects the arc length result. Run the calculator above with these values to get the exact updated output with step-by-step work.