What is the arc length?
Type the radius and the central angle, or any two of the radius, angle, arc length, and sector area. The arc length calculator finds the rest, with the chord.
- Arc length
- 10.471976
An arc with radius 10 and a central angle of 60° is 10.471976 long, and its sector has an area of 52.359878.
- Sector area
- 52.359878
- Chord length
- 10
- Sector perimeter
- 30.471976
Arc length: 10.471976. An arc with radius 10 and a central angle of 60° is 10.471976 long, and its sector has an area of 52.359878.
Arc length by central angle
How to calculate
Finds the arc length s = rθ of a circle from the radius and the central angle, or any two of the radius, angle, arc length, and sector area, with the chord.
Example with the default inputs (Radius 10, Central angle 60 °): An arc with radius 10 and a central angle of 60° is 10.471976 long, and its sector has an area of 52.359878.
Formula: s = rθ, where θ is the central angle in radians; the sector area is A = ½r²θ = ½sr.
- The central angle is more than 0° and at most 360°. It can be typed in degrees or radians; the formulas use radians (θ in radians = degrees × π ÷ 180).
- The radius, arc length, and sector area are positive numbers in any one unit; the area is in that unit squared.
- An arc longer than the circumference, or a sector bigger than the circle, has no answer.
Worked examples
Each example is checked against the calculator on every build.
- Radius 10, Central angle 60° gives Arc length 10.471976, Sector area 52.359878, Chord length 10, Sector perimeter 30.471976.Source: OpenStax, Precalculus 2e, §5.1 Angles (https://openstax.org/books/precalculus-2e/pages/5-1-angles)
- Radius 5, Arc length 12 gives Central angle 137.509871°, Sector area 30, Chord length 9.320391.
- Arc length 15, Sector area 60 gives Radius 8, Central angle 107.429587°, Chord length 12.897298.
- Central angle 90°, Sector area 314.159265 gives Radius 20, Arc length 31.415927, Chord length 28.284271.
- Radius 3, Central angle 360° gives Arc length 18.849556, Sector area 28.274334, Chord length 0.
How it works
An arc is part of a circle, cut off by two radii that meet at the centre at the central angle θ. With θ in radians:
- Arc length: s = rθ
- Sector area: A = ½r²θ, which also equals A = ½sr
- Chord length (the straight line between the ends of the arc): c = 2r sin(θ/2)
- Sector perimeter (the arc plus the two radii): P = s + 2r
Type any two of r, θ, s, and A. The calculator finds the other two with these rearrangements:
- r and θ: s = rθ, A = ½r²θ
- r and s: θ = s ÷ r, A = ½sr
- r and A: θ = 2A ÷ r², s = rθ
- θ and s: r = s ÷ θ
- θ and A: r = √(2A ÷ θ)
- s and A: r = 2A ÷ s, then θ = s ÷ r
An angle typed in degrees is converted to radians first: θ = degrees × π ÷ 180.
Assumptions
- The central angle is more than 0° and at most 360° (2π radians). A solved angle more than 360° has no answer: an arc longer than the circumference ("The arc is longer than the whole circle") or a sector bigger than the circle ("The sector is bigger than the whole circle").
- The radius, arc length, and sector area are positive numbers, all in one unit; the sector area is in that unit squared. The radius can be at most 10¹⁰⁰, the arc length at most 10¹⁰¹, and the sector area at most 10²⁰¹; a value typed or solved beyond these has no answer.
- At a full turn (360°, within one part in 10¹²) the two ends of the arc meet, so the chord is 0.
- Solved angles show in degrees. Answers show at most 6 decimals; a value below 0.0001 shows 6 significant figures. Rounding is half up on the decimal value.
- The chart draws the arc length against the central angle in radians, from 0 to 2π (360°), for the current radius.
Worked examples by hand
Radius 10, angle 60°. θ = 60 × π ÷ 180 = π/3 = 1.0472 rad. s = 10 × π/3 = 10.471976. A = ½ × 10² × π/3 = 52.359878. Chord = 2 × 10 × sin 30° = 10. Sector perimeter = 10.471976 + 20 = 30.471976.
Radius 5, arc 12. θ = 12 ÷ 5 = 2.4 rad (137.509871°). A = ½ × 12 × 5 = 30. Chord = 2 × 5 × sin 1.2 = 9.320391.
Arc 15, sector area 60. r = 2 × 60 ÷ 15 = 8. θ = 15 ÷ 8 = 1.875 rad. Chord = 16 × sin 0.9375 = 12.897298.
Angle 90°, sector area 100π. r = √(2 × 100π ÷ (π/2)) = √400 = 20. s = 20 × π/2 = 10π = 31.415927. Chord = 40 × sin 45° = 20√2 = 28.284271.
Radius 3, angle 360°. The arc is the whole circle: s = 2π × 3 = 18.849556, A = π × 3² = 28.274334, and the chord is 0.
Other questions people ask
What is the arc length formula?
s = rθ, where r is the radius and θ is the central angle in radians. With the angle in degrees, s = 2πr × (θ ÷ 360°). An arc of 60° on a circle of radius 10 is 10 × π/3 = 10.472 long.
How do I find the central angle from the arc length?
Divide the arc length by the radius: θ = s ÷ r, in radians. An arc of 12 on a circle of radius 5 has θ = 2.4 radians, which is 2.4 × 180° ÷ π = 137.51°.
How do I find the radius from the arc length and the angle?
r = s ÷ θ, with θ in radians. An arc of 31.4159 with a central angle of 90° (π/2 radians) has a radius of 31.4159 ÷ 1.5708 = 20.
What is the difference between arc length and chord length?
The arc follows the curve of the circle; the chord is the straight line between the two ends. The chord is always shorter: c = 2r sin(θ/2). For a 60° arc of radius 10, the arc is 10.472 and the chord is exactly 10.
How is the sector area related to the arc length?
The sector is the slice of the circle bounded by the arc and two radii. Its area is A = ½r²θ, which is also ½ × arc length × radius. That is why the calculator can find the radius from the arc length and the sector area: r = 2A ÷ s.
Why does the formula need radians?
A radian is defined so that an arc of one radius subtends an angle of 1 radian. That makes s = rθ exact with no conversion factor. The calculator accepts degrees and converts: radians = degrees × π ÷ 180.
Can the central angle be more than 360°?
No. An arc on one circle can go at most once around, so the angle is at most 360° (2π radians) and the arc is at most the circumference 2πr. An arc longer than that has no answer here.