What is the Arc Length from Degrees Calculator?
This calculator finds the length of a circular arc when you know the radius of the circle and the central angle measured in degrees. An arc is simply a portion of the circle's circumference, and the length of that arc is proportional to the fraction of the full 360° turn that the angle covers.
How to use it
Enter the radius (r) in any unit you like — centimeters, meters, inches, etc. Then enter the central angle in degrees (0 to 360). The calculator returns the arc length in the same unit as the radius, along with the angle converted to radians and the full circumference for reference.
The formula explained
The full circumference of a circle is \(2\pi r\). A central angle of \(\theta\) degrees covers \(\theta/360\) of the full circle, so the arc length is:
$$L = \frac{\theta}{360} \times 2\pi r$$
Because a full circle is 360°, dividing your angle by 360 gives the fraction of the circumference that the arc represents.
Worked example
Suppose \(r = 10\) and the central angle is 90° (a quarter circle). Then $$L = \frac{90}{360} \times 2\pi \times 10 = 0.25 \times 62.8319 = 15.708 \text{ units}.$$ The angle in radians is \(90 \times \frac{\pi}{180} = 1.5708\), and the full circumference is \(62.832\).
Common Arc Lengths by Angle
The table below uses a unit circle (radius \(r=1\)). Arc length is computed with \(L=\dfrac{\theta}{360}\times 2\pi r\). For any other radius, simply multiply the "as multiple of r" column by your radius.
| Angle (degrees) | Radians | Arc length (multiple of r) | Arc length (decimal, r=1) | Fraction of circle |
|---|---|---|---|---|
| 30° | \(\pi/6\) | \(\tfrac{\pi}{6}\,r\) | 0.5236 | 1/12 |
| 45° | \(\pi/4\) | \(\tfrac{\pi}{4}\,r\) | 0.7854 | 1/8 |
| 60° | \(\pi/3\) | \(\tfrac{\pi}{3}\,r\) | 1.0472 | 1/6 |
| 90° | \(\pi/2\) | \(\tfrac{\pi}{2}\,r\) | 1.5708 | 1/4 |
| 120° | \(2\pi/3\) | \(\tfrac{2\pi}{3}\,r\) | 2.0944 | 1/3 |
| 180° | \(\pi\) | \(\pi\,r\) | 3.1416 | 1/2 |
| 270° | \(3\pi/2\) | \(\tfrac{3\pi}{2}\,r\) | 4.7124 | 3/4 |
| 360° | \(2\pi\) | \(2\pi\,r\) | 6.2832 | 1 (full circle) |
Key Terms
- Arc — a continuous portion of the circle's edge (circumference). Its length \(L\) is what this calculator finds from the radius and central angle.
- Central angle (θ) — the angle, measured in degrees here, formed at the center of the circle by the two radii that bound the arc. A larger \(\theta\) sweeps a longer arc; at 360° the arc becomes the whole circumference.
- Radius (r) — the distance from the center to any point on the circle. Arc length scales directly with \(r\): double the radius and the arc for the same angle doubles.
- Radian — the angle that subtends an arc equal in length to the radius. Because \(360^\circ = 2\pi\) radians, converting to radians gives the compact form \(L = r\theta_{\text{rad}}\).
- Circumference — the arc length of the full circle, \(C = 2\pi r\). Every arc length is a fraction \(\theta/360\) of this value.
- Chord — the straight line joining the two endpoints of the arc. It is always shorter than the arc it spans and is not the same as arc length.
- Sector — the "pie-slice" region bounded by the arc and its two radii. The arc is its curved boundary; its area is \(\tfrac{\theta}{360}\pi r^2\).
FAQ
What unit is the arc length in? The same unit as the radius you entered. If r is in meters, the arc length is in meters.
Can the angle be more than 360°? This tool limits the angle to 0–360°. For angles beyond a full turn, subtract multiples of 360° first.
How do I get the chord length instead? The chord (straight line between the arc's endpoints) is \(2r \times \sin(\theta/2)\), which differs from the curved arc length.
