Measuring a curve
An arc is a piece of a circle’s edge, cut off by two radii that meet at the centre. Its length depends on two things: how big the circle is and how wide the angle is. Arcs turn up in curved paths and garden beds, bent pipes and rails, clock faces, wheel turns and geometry exercises.
This arc length calculator gives the arc length from a radius and an angle in degrees or radians, along with the area of the slice (the sector) and the straight line between the two ends (the chord). The second mode runs backwards: give it the radius and the arc, and it finds the angle.
How to use it
- Choose From radius and angle, or Angle from an arc.
- Enter the radius: the distance from the centre to the edge.
- Enter the central angle in degrees or radians, or the arc length.
- Read the arc length or angle, then the sector area, the chord and the share of the full circle.
Keep the units the same: a radius in metres gives an arc in metres and a sector area in square metres.
The arc length formula
arc length s = r × θ (θ in radians)
in degrees: s = 2πr × angle ÷ 360
sector area = ½ × r² × θ
chord = 2r × sin(θ ÷ 2)A radian is the angle that makes an arc exactly one radius long, which is why the formula is so simple in radians. A full turn is 2π radians, or 360°, and its arc is the whole circumference, 2πr.
Worked example: radius 10, angle 60°
- In radians: 60 × π ÷ 180 = 1.0472
- Arc length: 10 × 1.0472 = 10.472
- Sector area: ½ × 10² × 1.0472 = 52.3599
- Chord: 2 × 10 × sin(30°) = 10
A 60° slice of a circle with radius 10 has a curved edge of 10.472, an area of 52.3599 and a straight edge of exactly 10. The chord equals the radius here because the slice is an equilateral triangle with a bulge.
Arc lengths for radius 10
| Angle | Radians | Arc length (r = 10) |
|---|---|---|
| 30° | 0.5236 | 5.236 |
| 45° | 0.7854 | 7.854 |
| 60° | 1.0472 | 10.472 |
| 90° | 1.5708 | 15.708 |
| 180° | 3.1416 | 31.4159 |
| 360° | 6.2832 | 62.8319 |
For any other radius, scale the last column: a radius of 5 halves each arc, and a radius of 20 doubles it. The radians column never changes, because it depends only on the angle.
Related tools
- For a whole circle, the circle calculator gives circumference and area.
- For triangles and trig values, use the triangle and trigonometry calculator.
- For curved beds and paths, measure the arc and the radius, then work out the area with the sector line above.
Frequently asked questions
How do I find the length of an arc?
Multiply the radius by the central angle in radians. If the angle is in degrees, multiply the circumference, 2πr, by the angle divided by 360. For a radius of 10 and an angle of 90°, that is 62.83 × 0.25, which gives an arc of about 15.71.
How do I convert degrees to radians?
Multiply the degrees by π and divide by 180. So 90° is π ÷ 2, about 1.5708 radians, and 180° is π, about 3.1416. To go back, multiply radians by 180 and divide by π. The calculator shows the angle in both units with every answer.
What is the difference between arc length and chord length?
The arc follows the curve of the circle between two points, while the chord is the straight line joining them. The arc is always the longer of the two. For small angles they are almost the same; for a half circle, the arc is about 1.57 times the chord.
How do I find the angle from an arc length?
Divide the arc length by the radius to get the angle in radians, then multiply by 180 and divide by π for degrees. An arc of 15 on a radius of 10 is 1.5 radians, about 85.94°. Choose Angle from an arc above to do this directly.
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