How to use it
- Enter the radius.
- Enter the central angle.
- Choose degrees or radians.
- Read the arc length, sector area, chord, segment and arc height.
Arc length formula
An arc is part of a circle's circumference. Its length depends on the radius r and the central angle θ:
s = r × θ (θ in radians)
s = 2πr × θ / 360° (θ in degrees)
Example: r = 10 in, θ = 60°.
- θ in radians = 60 × π / 180 ≈ 1.0472
- s = 10 × 1.0472 ≈ 10.47 in
Choose degrees or radians and the calculator converts for you. Angles from just above 0 up to a full 360° (2π) work. Everything is computed in your browser.
All the results
| Result | Formula | r = 10, θ = 60° |
|---|---|---|
| Arc length | s = rθ | 10.47 |
| Sector area | A = r²θ / 2 | 52.36 |
| Chord length | c = 2r sin(θ/2) | 10.00 |
| Segment area | r²(θ − sin θ) / 2 | 9.06 |
| Arc height (sagitta) | h = r(1 − cos(θ/2)) | 1.34 |
The sector is the pie slice between two radii. The segment is the region between the chord and the arc, the sector minus the triangle. The sagitta is the height of the arc above the chord.
Practical examples
- Curved edging: a garden bed curves along a quarter circle with an 8 ft radius. Edging needed: 2π × 8 / 4 ≈ 12.57 ft.
- Arched window or doorway: for an arch spanning a 36-inch opening with a 6-inch rise, the chord and sagitta describe the arch; the calculator helps you check a chosen radius and angle against those measurements.
- Track and road curves: the length of a curve with a known radius and turning angle.
- Pie charts: a slice representing 25% has a 90° central angle.
Degrees vs. radians
A radian is the angle at which the arc length equals the radius, so the formula s = rθ is simplest in radians. One full circle is 2π radians (about 6.2832), so 1 radian ≈ 57.2958°.
| Degrees | Radians |
|---|---|
| 30° | π/6 ≈ 0.5236 |
| 45° | π/4 ≈ 0.7854 |
| 90° | π/2 ≈ 1.5708 |
| 180° | π ≈ 3.1416 |
| 360° | 2π ≈ 6.2832 |
For more angle conversions, use the trig calculator.
Common mistakes
- Using degrees in s = rθ. That formula needs radians; with degrees, use s = 2πr × θ / 360.
- Confusing arc length with chord length. The arc follows the curve; the chord is the straight line between the endpoints and is always shorter.
- Diameter instead of radius. Halve a measurement taken across the full circle.
- Angles over 180°. For a major arc, the segment is larger than half the circle and the chord sits on the far side of the center; the formulas still work. The calculator accepts any angle up to a full 360°.
For full circles, circumference and area, use the circle calculator.
Working backward
Sometimes you know the arc length and need the angle or radius:
- Angle from arc and radius: θ (radians) = s / r. An arc of 15 ft on a 10 ft radius spans 1.5 rad ≈ 85.94°.
- Radius from arc and angle: r = s / θ. A 20 ft curve turning 45° (0.7854 rad) has a radius of about 25.46 ft.
Plug the result back into the calculator to see the sector area, chord and arc height for that shape.
Frequently asked questions
How do I find arc length?
Multiply the radius by the angle in radians, or use s = 2πr × θ / 360 with degrees.
What is the difference between a sector and a segment?
A sector is the pie slice between two radii; a segment is the area between the chord and the arc.
What is the sagitta?
The height of the arc above the chord: h = r(1 − cos(θ/2)).
Can I enter the angle in radians?
Yes. Choose radians as the angle unit.
What is the arc length of a semicircle?
Half the circumference, πr. For a 10-inch radius, about 31.42 inches.
Last updated: November 12, 2026