What is arc length?
The arc length is the distance measured along the curved path of a circle between two points on its circumference. When you draw two radii they cut out a curved portion called an arc. The length of that curve — not the straight-line chord connecting the endpoints — is the arc length.
Arc length appears in dozens of practical contexts: the length of a curved road, the travel distance of a rotating wheel tooth, a pipeline running along a curved route, the reach of a robotic arm, or cable wrap around a drum. Whenever geometry involves a circular path or a rotating system, arc length is the key measurement.
The core formula
The fundamental relationship is elegantly simple when the central angle θ is in radians:
s = r × θ (θ in radians)
s = r × θ × π/180 (θ in degrees)Where s is the arc length, r is the radius, and θ is the central angle. Because a full circle spans 2π radians and the full circumference is 2πr, this formula is a proportional slice: s = (θ / 2π) × 2πr.
Complete arc geometry at a glance
Arc length is one member of a family of related measurements. This calculator computes all of them simultaneously:
Radius (r)
Distance from the circle center to any point on the circumference.
Central Angle (θ)
The angle at the center between the two radii. Supported in degrees, radians, turns, and gradians.
Arc Length (s)
s = r θThe curved distance between the two endpoints along the circle.
Chord Length (c)
c = 2r sin(θ/2)The straight-line distance connecting the two arc endpoints.
Sagitta / Height (h)
h = r (1 − cos(θ/2))The perpendicular height from the chord midpoint up to the arc midpoint.
Apothem (a)
a = r cos(θ/2)Distance from the center to the chord. Apothem and sagitta together equal the radius.
Sector Area
A = ½ r² θThe pie-slice region bounded by two radii and the arc.
Segment Area
A = ½ r² (θ − sin θ)The region between the chord and the arc only — sector minus the central triangle.
Solving from any two known values
You often know only two of the five primary values — radius, angle, arc length, chord, or sagitta. This calculator handles all ten input pairs:
r and θDirect: s = r θ, then derive c, h, and areas.
r and sθ = s / r, then derive c, h, and areas.
r and cθ = 2 arcsin(c / 2r) — requires c ≤ 2r.
r and hθ = 2 arccos(1 − h/r) — requires h < 2r.
s and θr = s / θ, then compute all other values.
c and θr = c / (2 sin(θ/2)), angle drives the arc.
h and θr = h / (1 − cos(θ/2)), then all others follow.
s and cSolved numerically (bisection): finds θ such that 2(s/θ) sin(θ/2) = c.
s and hSolved numerically (bisection): finds θ from (s/θ)(1 − cos(θ/2)) = h.
c and hDirect: r = (c² + 4h²) / 8h, then θ = 2 arcsin(c / 2r).
Supported angle units
Angles can be expressed in four common units. The calculator converts internally to radians before computing:
Degrees (°)
Most familiar unit. Full circle = 360°.
rad = deg × π/180Radians (rad)
Natural mathematical unit. Full circle = 2π ≈ 6.2832 rad.
Turns
One turn = one full revolution. Full circle = 1 turn.
rad = turns × 2πGradians (gon)
Used in surveying. Full circle = 400 grad.
rad = grad × π/200Multi-turn angles and normalization
Angles greater than 360° (one full revolution) are fully supported. A 2.5-turn angle represents an arc 2.5 times the full circumference — useful for coils, spiral springs, or winding drum calculations. The calculator reports three related values:
Raw θThe exact angle you entered, used for all area and arc computations.
Number of turnsθ / (2π) — shows how many times the arc wraps around the circle.
Normalized θThe remainder angle in [0°, 360°) or [0, 2π), useful for geometric reference.
Practical examples
Example 1 — Quarter circle (r = 10, θ = 90°)
Given: r = 10, θ = 90° = π/2 rad
s = r × θ = 10 × π/2 ≈ 15.7080
c = 2 × 10 × sin(45°) ≈ 14.1421
h = 10 × (1 − cos(45°)) ≈ 2.9289
Sector area = ½ × 100 × π/2 ≈ 78.5398
Segment area ≈ 28.5398
Fraction of circle = 25%Example 2 — From chord and sagitta (c = 8, h = 2)
Given: c = 8, h = 2
r = (c² + 4h²) / (8h) = (64 + 16) / 16 = 5
θ = 2 × arcsin(8 / 10) ≈ 106.26° ≈ 1.8546 rad
s = 5 × 1.8546 ≈ 9.2730
Sector area ≈ 23.1825
Segment area ≈ 15.1825Sector vs. segment area
Sector area
The full pie-slice region enclosed by two radii and the arc. Includes the triangular portion near the center.
A_sector = ½ r² θSegment area
Only the curved cap region between the chord and the arc — the sector minus the isosceles triangle formed by the two radii and the chord.
A_seg = ½ r² (θ − sin θ)Common applications
Engineering & machining
Calculate material length for curved brackets, bent pipes, or wire routing around pulleys.
Architecture & construction
Determine the length of curved walls, arched windows, or circular staircases.
Robotics & CNC
Compute the travel distance of end-effectors or cutting tools on circular paths.
Astronomy
Estimate the arc distance between two objects on the celestial sphere from an angular separation.
Road & rail design
Plan curve lengths in highway or railway alignment where the radius and deflection angle are specified.
Sports & tracks
Calculate the length of circular track lanes or the arc of a discus throw sector.
Tips for accurate results
Consistent units
All length inputs (r, s, c, h) must be in the same unit. The calculator is unit-agnostic — results are in whatever unit you use for input.
Minor vs. major arc
A chord divides a circle into two arcs. When solving from chord alone (without an angle), the calculator returns the minor arc (θ ≤ π). To obtain the major arc, supply an angle greater than 180°.
Physical constraints
The chord can never exceed the diameter (c ≤ 2r), and the sagitta must be less than 2r. The sagitta for a minor arc is always less than or equal to r.
Precision
Real-world measurements seldom need more than 4–6 significant figures. Use 8 only when verifying symbolic computations.