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Arc Length Calculator

Calculate arc length, chord length, sector area, segment area, and sagitta

Arc Length Calculator

Calculate From

Radius (r)

Angle (degrees)

Understanding Arcs

An arc is a portion of a circle's circumference. Arc length depends on the radius and the central angle. The arc, along with two radii, forms a sector (like a pizza slice), while the arc and chord form a segment.

Key Terms:

Arc: The curved portion of the circle's edge
Chord: Straight line connecting the arc's endpoints
Sector: Pizza slice area (arc + two radii)
Segment: Area between arc and chord
Sagitta: Height from chord midpoint to arc

Arc Formulas

s = rθ

Arc length (radians)

c = 2r sin(θ/2)

Chord length

A = (1/2)r²θ

Sector area

h = r(1 - cos(θ/2))

Sagitta

How to Use This Arc Length Calculator

Choose which pair of measurements you know — radius and angle (degrees or radians), arc length and radius, arc length and angle, or chord length and radius — and the calculator solves the rest: arc length, chord length, sagitta (arc height), sector area, and segment area. Because every quantity of a circular arc is determined by just two independent values, one pair is always enough.

Worked Example: A 90° Arc of Radius 5 m

Convert the angle to radians: θ = 90 × π/180 = π/2 ≈ 1.5708 rad. The arc length is s = rθ = 5 × 1.5708 ≈ 7.854 m — exactly a quarter of the circumference (2πr ≈ 31.42 m). The chord connecting the endpoints is c = 2r sin(θ/2) = 10 × sin(45°) ≈ 7.071 m, and the sagitta — the bulge height from chord to arc — is h = r(1 − cos(θ/2)) = 5 × (1 − 0.7071) ≈ 1.464 m.

Finding an Arc from a Chord: The Sagitta Method

In carpentry and masonry you often can't measure the radius directly — only the chord (span) and the rise of a curved piece. From the chord-and-radius mode you can work backwards, and the classic relation r = (c²/8h) + (h/2) recovers the radius of an arch from its span c and rise h. A doorway arch spanning 90 cm with a 10 cm rise, for instance, comes from a circle of radius (8100/80) + 5 ≈ 106.25 cm.

Common Applications

Construction: laying out arched windows, doorways, and curved stairs.
Fabrication: cutting curved trim, bending pipe or rebar to a specified radius.
Road and rail design: curve lengths for alignment and signage placement.
Geometry homework: checking s = rθ, chord, and segment-area answers step by step.

Frequently Asked Questions

What is the formula for arc length?+

Arc length equals radius times central angle in radians: s = rθ. With degrees, use s = (θ/360) × 2πr. For example, a 120° arc of a circle with radius 9 has length (120/360) × 2π × 9 ≈ 18.85 units, one third of the full circumference.

What is the sagitta of an arc?+

The sagitta (arc height) is the perpendicular distance from the midpoint of the chord to the arc itself, computed as h = r(1 − cos(θ/2)). Builders use it to describe the rise of an arch, and together with the chord it uniquely determines the circle's radius.

Can I find the angle if I know the arc length and radius?+

Yes. Select the arc-and-radius input mode and the calculator solves θ = s/r, reporting the angle in both radians and degrees. It then derives the chord, sagitta, sector area, and segment area from the same two inputs.

What is the difference between arc length and chord length?+

The arc is the curved path along the circle between two points, while the chord is the straight line joining them, so the chord is always shorter. For small angles they are nearly equal, but they diverge as the angle grows: at 180° the arc is πr ≈ 3.14r while the chord is only 2r.

Is this arc length calculator free and does it work on mobile?+

Yes. The tool is free, needs no account, and runs entirely in your browser — nothing you enter is sent anywhere. It works on phones and tablets, which makes it practical for on-site measurements in the workshop or on a job site.

Arc Length Calculator