Arc Length Calculator
Find the arc length of a circle from the radius or diameter and the central angle (degrees, radians or gradians) — or from the chord, the arc height, the sector area or the arc itself. Also gives the exact π form, sector and segment area, chord, sagitta, perimeter and worked steps.
How it works
θ (radians) = angle° × π ÷ 180 · arc s = r × θ · sector area = ½ r²θ · chord c = 2r × sin(θ ÷ 2) · arc height h = r(1 − cos(θ ÷ 2)) · segment area = ½ r²(θ − sin θ) from a chord: θ = 2 asin(c ÷ 2r) · from chord and height: r = c² ÷ 8h + h ÷ 2 · from the arc: θ = s ÷ r · from the sector area: r = √(2A ÷ θ) or θ = 2A ÷ r²
Once the central angle is in radians, the arc length is the radius times that angle, s = rθ. That simple rule is the reason radians exist. Enter the angle in degrees, radians or gradians, and use the diameter if that is what you measured. The sector (the pie-slice area) is ½r²θ. The chord, the straight line between the ends of the arc, comes from the isosceles triangle formed by the two radii: c = 2r·sin(θ ÷ 2). The arc height, or sagitta, is how far the arc bulges above the chord. The segment is the area between the chord and the arc: the sector minus that triangle. You can also work backwards. From a radius and a chord, the angle is 2·asin(c ÷ 2r), with a choice of the minor or major arc. From a chord and an arc height, which is how arches, curved walls and bent pipes are usually measured, the radius is c² ÷ 8h + h ÷ 2. An arc length or a sector area plus the radius also gives the angle. With a whole-number radius and angle in degrees, the calculator also gives the exact answer in terms of π.
Worked example
Radius 10, central angle 60°: θ = 60 × π ÷ 180 ≈ 1.047198 rad. The arc length is 10 × 1.047198 ≈ 10.4720, or exactly 10π/3. The sector area is ½ × 100 × 1.047198 ≈ 52.3599 (50π/3). The chord is 2 × 10 × sin 30° = 10 exactly, the arc height is 10 × (1 − cos 30°) ≈ 1.3397, and the segment area is ½ × 100 × (1.047198 − 0.866025) ≈ 9.0586. Working backwards, a chord of 10 with an arc height of 1.34 gives a radius of 100 ÷ 10.72 + 0.67 ≈ 10.0, so the same 60° arc.
Sources: Circular sector — Wikipedia · Circular segment — Wikipedia
Frequently asked questions
Why convert the angle to radians first?
The tidy formula s = rθ only holds when θ is in radians. In degrees you would write s = 2πr × (angle ÷ 360), which is the same thing with the π ÷ 180 conversion folded in. You can type the angle in radians or gradians with the unit menu.
What’s the difference between the arc and the chord?
The arc is the curved distance along the circle, and the chord is the straight shortcut between the same two endpoints. The chord is always shorter, and the two get closer as the angle shrinks toward zero.
How do I find the arc length without the angle?
Use the radius and the chord: θ = 2·asin(c ÷ 2r), then s = rθ. If you only have the chord and the arc height, find the radius first with r = c² ÷ 8h + h ÷ 2. Choose the matching option under “What do you know?” to have both done for you.
How do I find the sector area as a fraction of the circle?
A sector is the same fraction of the circle’s area as its angle is of 360°. A 60° sector is 60 ÷ 360 = 1/6 of πr². For radius 10 that is 1/6 × 100π ≈ 52.36, matching the ½r²θ result.
What is the segment area?
It is the area between the chord and the arc: the sector minus the triangle formed by the two radii and the chord, ½r²(θ − sin θ). For radius 10 and 60° it is about 9.06.
What is a major arc?
Two points on a circle split it into two arcs. The shorter one is the minor arc and the longer one, more than 180°, is the major arc. Both share the same chord. With a radius and a chord, pick which one you want.
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This calculator is for educational purposes. Double-check important results before acting on them.