Arc length calculator — arc, sector area, chord and segment from r and θ

How to use the arc length calculator

1. Type the Radius of the circle.
2. Type the Central angle, the angle at the centre between the two radii.
3. Choose the Angle unit: Degrees or Radians.
4. Read Arc length, then Sector area, Chord length, Segment area and the angle in both units. The Formulas box shows the working.

What it does and when to use it

An arc is part of a circle’s edge. Its length matters when you bend a strip of metal or wood around a curve, mark out a curved path or flower bed, or work out how far a point on a wheel travels as it turns. The sector area tells you how much material a fan-shaped piece needs, and the chord gives the straight-line gap between the ends.

For a whole circle, the circle calculator is simpler. The perimeter calculator adds the two straight radii to the arc for a sector’s full outline. To switch between degrees and radians first, use the angle converter.

How it works

The tool converts the angle to radians (θ = degrees × π/180), then applies:

• Arc length: s = r·θ
• Sector area: A = ½·r²·θ
• Chord length: chord = 2r·sin(θ/2)
• Segment area: ½·r²·(θ − sin θ)

Worked examples

Step by step. Radius 12, angle 45°.

1. Convert to radians: 45 × π/180 = 0.7853981634.
2. Multiply by the radius: 12 × 0.7853981634 = 9.424777961.

The Formulas box shows s = r·θ = 12 × 0.7853981634 = 9.424777961. The sector area is 56.54866776 and the chord is 9.184402377, a little shorter than the arc.

Angle already in radians. Radius 5 and 2.5 rad: no conversion is needed, so s = 5 × 2.5 = 12.5 and Sector area: 31.25 (½ × 25 × 2.5). The tool also reports the angle as 143.2394488°.

A half circle. Radius 8 and 180°: the arc is 25.13274123 (half of the circumference) and Chord length: 16, the diameter. Here the segment and sector areas are equal, both 100.5309649, because the chord passes through the centre.

Limits and tips

• Radius and angle must be greater than zero.
• Angles larger than one full turn are rejected rather than wrapped around.
• Pick the correct Angle unit. Entering 60 with Radians selected means about 3,438°, which is refused.
• Results show up to 10 significant figures and are calculated in your browser.