Arc length calculator, from radius and angle.

Arc length is s = r × θ, with the angle θ in radians. In degrees the same rule reads s = 2 × π × r × (angle ÷ 360). Take a radius of 10 cm and an angle of 60°. The angle in radians is 60 × π ÷ 180 = 1.0472. Then s = 10 × 1.0472 = 10.4720 cm. As a check, 60° is one sixth of the circle and the circumference is 2 × π × 10 = 62.8319 cm, and 62.8319 ÷ 6 = 10.4720 cm.

How to use the converter

1. Choose Find: the arc length from the angle, or the angle from an arc length.
2. Pick Radius or Diameter, type it, then type the Angle and choose Degrees or Radians.
3. Read the big result and The working. The cards give the sector area, the chord, the share of the circle and the angle in the other unit.
4. Use Copy result or Copy link to keep the answer.

Why the angle must be in radians

A radian is the angle where the arc is as long as the radius. So an angle of θ radians cuts an arc of θ radii, which is r × θ. If you type degrees into s = r × θ, the answer is wrong by a factor of 57.30: 60 as an angle gives 10 × 60 = 600 cm, against the true 10.47 cm. The tool changes degrees to radians for you, and NIST Special Publication 811, Appendix B.8 lists the degree as 1.745 329 E-02 radian (checked 24 Oct 2026).

Other things you get from the same two numbers

The sector is the slice between the two radii and the arc. Its area is 1/2 × r² × θ. For r = 10 and θ = 60° that is 0.5 × 100 × 1.0472 = 52.360 cm². The chord is the straight line between the ends of the arc: 2 × r × sin(θ ÷ 2). For 60° that is 2 × 10 × sin 30° = 10.000 cm, the same as the radius, because the triangle is equilateral.

Arc length for radius 10 at common angles

Worked examples

• A curved edge, diameter 40 cm, angle 90°. The radius is 20. s = 20 × 1.5708 = 31.416 cm.
• An angle in radians. r = 3 m, θ = 1.2 rad. s = 3 × 1.2 = 3.6 m.
• The angle from an arc. r = 10 cm and an arc of 15 cm. θ = 15 ÷ 10 = 1.5 rad, which is 1.5 × 180 ÷ π = 85.944°. Use the "Angle from the arc length" mode.
• A full turn. 360° is 2π radians, so s = 2πr, the circumference. For r = 10 that is 62.832 cm.

Segment area and the bit between arc and chord

A segment is the cap between an arc and its chord. Its area is the sector minus the triangle: 1/2 × r² × (θ − sin θ), with θ in radians. For r = 10 and 60° (θ = 1.0472), the sector is 52.3599, the triangle is 1/2 × 100 × sin 60° = 43.3013, and the segment is 52.3599 − 43.3013 = 9.0586. Use the same unit for the radius throughout, and the area is in square units.

Arc length from the chord and the rise

Often you cannot measure the angle or the radius, only the straight chord c and the rise h at the middle (the distance from the chord to the arc). The radius is r = c² ÷ (8h) + h ÷ 2. For c = 24 and h = 5: r = 576 ÷ 40 + 2.5 = 16.9. The angle is θ = 2 × asin(c ÷ (2r)) = 2 × asin(12 ÷ 16.9) = 1.5792 radians, which is 90.48°. The arc is r × θ = 16.9 × 1.5792 = 26.6879. Type the radius 16.9 and the angle into the tool to see the same result, and the chord card will show 24.

Length of one degree

One degree of arc is r × π ÷ 180, about 0.017453 × r. Multiply that by the number of degrees. For r = 50 and 37°: 37 × 0.8727 = 32.2886.

Share of the circumference

An arc is the same share of the circumference as its angle is of 360°. A quarter of a circle of radius 10 is 62.8319 ÷ 4 = 15.708. A twelfth is 62.8319 ÷ 12 = 5.236. This is another way to check any answer: the arc must never be more than the whole circumference for an angle under 360°, and always more than the chord.

Mistakes to avoid

• Degrees in the radian formula. This is the most common error and gives an answer about 57 times too large.
• Using the diameter as the radius. Doubles the arc. Choose Diameter in the tool when you have it.
• A calculator in the wrong mode. If you work out sin or the radians yourself, check that it is set to degrees or radians as the angle is written.
• Mixing arc and chord. The chord is the straight line, shorter than the arc.

To change an angle between units, use degrees to radians. The unit circle chart lists the common angles.