Last updated: 2026-05-19

Arc length

Arc length is the distance measured along the curved edge of a circular arc — the portion of a circle's circumference between two points. It depends on two quantities: the radius r of the circle and the central angle θ subtended at the centre.

This calculator computes three related quantities from r and θ:

• Arc length s — the curved distance along the circle
• Chord length c — the straight-line distance between the two endpoints
• Fraction of circumference f — what share of the full circle the arc represents

The arc length formula

The arc length formula is:

$s = r \cdot \theta$

where θ is in radians. The formula is a direct consequence of how the radian is defined: one radian is the angle that subtends an arc whose length equals the radius. So an arc spanning θ radians on a circle of radius r has length r × θ.

Why the formula works

The full circumference is 2πr (at θ = 2π radians). For a fraction θ/(2π) of the circle, the arc is:

$s = 2\pi r \cdot \frac{\theta}{2\pi} = r\theta$

This confirms the formula and also shows that the "Fraction of Circumference" output f = θ/(2π).

Working with degrees

Angles are commonly expressed in degrees, but the formula requires radians. The conversion is:

$\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}$

This calculator accepts degrees directly — selecting the ° unit on the angle field converts automatically. The equivalent hand calculation is:

$s = 2\pi r \times \frac{\theta_{\text{deg}}}{360}$

Example: r = 8 m, θ = 135°

• Convert: 135° × π/180 = 3π/4 ≈ 2.3562 rad
• Arc length: s = 8 × 2.3562 ≈ 18.85 m
• As a fraction of the circumference: 135/360 = 0.375 (three-eighths of the circle)

Chord length

The chord c is the straight line joining the two endpoints of the arc. It is always shorter than the arc (unless the angle approaches zero, in which case they converge):

$c = 2r \sin\!\left(\frac{\theta}{2}\right)$

Worked example: r = 10 m, θ = 90° (π/2 rad)

Quantity	Formula	Value
Arc length s	r × π/2	10 × 1.5708 ≈ 15.71 m
Chord c	2 × 10 × sin(45°)	20 × 0.7071 ≈ 14.14 m
Fraction f	90 / 360	0.25

The arc is about 11% longer than the chord for this 90° case. For small angles, the two are nearly equal; for a 180° arc, the chord equals the diameter (2r) while the arc is πr.

Special cases

Central angle	Arc length	Chord	Notes
0°	0	0	Degenerate point
60°	πr/3	r	Equilateral triangle: chord = radius
90°	πr/2	$r\sqrt{2}$	Quarter circle
180°	πr	2r	Chord = diameter
360°	2πr	0	Full circumference; endpoints meet

The 60° case is notable: a chord spanning exactly 60° has the same length as the radius. Six such chords tile the circle exactly, which is why a regular hexagon fits perfectly around a circle.

Where arc length is used

Arc length appears wherever a measurement follows a circular path rather than a straight line: the running length of a curved road or rail section, the material needed to bend a pipe or sheet through a given angle, the distance a point travels on a rotating wheel or gear, and the layout of arcs in architecture and drafting. In each case the radius and the central angle are the two inputs, and the curved distance s = r·θ is the quantity of interest. The chord length and fraction of circumference give the complementary straight-line and proportional views of the same arc.

Frequently Asked Questions (FAQ)

What is the arc length formula?

The arc length s of a circular arc is s = r·θ, where r is the radius and θ is the central angle in radians. For a circle of radius 5 m with a central angle of 2 radians, s = 5 × 2 = 10 m. The formula follows directly from the definition of the radian: one radian is the angle that subtends an arc equal in length to the radius.

How do I calculate arc length in degrees?

Convert the angle to radians first: θ_rad = θ_deg × π / 180, then apply s = r·θ_rad. Equivalently, s = 2πr × (θ_deg / 360) — the arc is that fraction of the full circumference. For r = 10 m and θ = 90°: s = 2π × 10 × (90/360) = 2π × 10 × 0.25 = 5π ≈ 15.71 m. This calculator accepts degrees directly — no manual conversion needed.

What is the difference between arc length and chord length?

The arc length s = r·θ is the curved distance along the circle between two points, while the chord c = 2r·sin(θ/2) is the straight-line distance connecting those same two points. The chord is always shorter than the arc (or equal when θ → 0).

For a 90° arc on a 10 m radius circle: arc ≈ 15.71 m, chord = 10√2 ≈ 14.14 m. The difference grows with the central angle — at 180° the chord equals the diameter but the arc is half the circumference (πr vs 2r).

How is arc length related to circumference?

The circumference C = 2πr is the arc length of the full circle (θ = 2π radians = 360°). Any arc is a fraction of the circumference: s = C × (θ / 2π). The "Fraction of Circumference" output (f = θ / 2π) shows this directly — f = 0.25 means the arc spans a quarter of the circle. At θ = 2π the arc equals the circumference exactly.