Arc length calculator

How do I find the length of an arc?

Arc length = radius × angle in radians, or 2πr × degrees ÷ 360. A 90° arc of a circle with radius 10 is 2π × 10 ÷ 4 = 15.71.

How do I find the area of a sector?

½ × r² × angle in radians, or πr² × degrees ÷ 360: a 90° sector with radius 10 is 78.54.

What is the chord length?

The straight line between the ends of the arc: 2r × sin(angle ÷ 2). For 90° and radius 10 it is 14.14.

What is a circular segment?

The area between the chord and the arc: the sector minus the triangle. For the example, 78.54 − 50 = 28.54.

How do I convert degrees to radians?

Multiply by π ÷ 180: 90° is π/2 = 1.5708 radians. The degrees to radians converter does it for any angle.

How do I find the radius or the angle when I know the arc length?

Turn s = rθ around: r = s ÷ θ, with θ in radians. An 8 ft arc that spans 60° (1.0472 radians) belongs to a circle of radius 8 ÷ 1.0472 = 7.64 ft; enter that radius and 60° and the arc comes back as 8.00 ft.

Does it work for angles above 180°?

Yes, up to a full circle: 360° gives the circumference 2πr and the area πr². The chord is longest at 180° (the diameter) and shortens again afterwards. Past 180° the sine term in the segment formula is negative, so the triangle is added to the sector instead of subtracted. For a 270° arc of radius 10 the arc is 47.12, the chord 14.14 and the segment 285.62.

The formulas and what they assume

A radian is defined as arc length divided by radius, so s = rθ (OpenStax Precalculus 2e, section 5.1). A radian is a ratio of two lengths and has no unit, so the arc comes out in the unit of the radius. The sector is the same fraction of the disk as the arc is of the circle: ½θr². The chord is 2r·sin(θ ÷ 2), and the circular segment, sector minus the triangle of the chord and two radii, is ½r²(θ − sin θ) (Wolfram MathWorld, “Circular Segment”). The calculator takes degrees, converts with θ = degrees × π ÷ 180, and accepts angles above 0° up to 360°.

Worked example: a lawn sprinkler

OpenStax Example 9 has a sprinkler that reaches 20 feet and sweeps 30°, or 0.5236 radians. The water edge is an arc of 20 × 0.5236 = 10.47 ft, and the watered sector is ½ × 0.5236 × 20² = 104.72 ft², which matches the book’s 104.72 square feet. The chord is 10.35 ft and the segment between chord and arc 4.72 ft².

Common mistakes

Putting degrees into a formula that wants radians: r = 20 and 30° typed into s = rθ gives 600, not 10.47. The opposite slip is typing a radian value such as 0.5236 into the degrees box, which reads as half a degree. Using the diameter as the radius doubles the arc and quadruples the sector area. And a chord is not an arc: at 30° it is 1.1% shorter, at 90° already 10.0%, so a curved edge needs the arc length.

Sources: OpenStax, Precalculus 2e, section 5.1 “Angles” (arc length, area of a sector, Example 9); Wolfram MathWorld, “Circular Segment” (chord and segment formulas). Related tools: circle calculator, degrees to radians converter, polygon calculator and trigonometry calculator. How we check our calculators.