Calculate arc length, radius, or central angle from two values with unit conversions for meters, inches, feet, cm, km, degrees, or radians.
Arc Length Formula
The arc length formula relates the radius of a circle, the central angle, and the length of the curved arc between the angle’s endpoints.
L = r\theta
- L = arc length
- r = radius
- θ = central angle in radians
If the central angle is entered in degrees, the calculator first converts it to radians. The degree version of the arc length formula is:
L = r * \frac{\pi\theta}{180}- L = arc length
- r = radius
- θ = central angle in degrees
- π = pi, approximately 3.14159
To solve for radius, the formula is rearranged as:
r = \frac{L}{\theta}To solve for the central angle in radians, the formula is rearranged as:
\theta = \frac{L}{r}The calculator has three main functions:
- Find arc length: enter the radius and central angle. The calculator uses L = rθ.
- Find radius: enter the arc length and central angle. The calculator uses r = L / θ.
- Find central angle: enter the radius and arc length. The calculator uses θ = L / r.
Length units are converted internally to meters, and angle values are converted internally to radians before the calculation is made.
Common Angle Conversions for Arc Length
Arc length calculations use radians. If your angle is in degrees, convert it to radians before using the basic formula by hand.
| Degrees | Radians | Fraction of Full Circle |
|---|---|---|
| 30° | π/6 | 1/12 |
| 45° | π/4 | 1/8 |
| 60° | π/3 | 1/6 |
| 90° | π/2 | 1/4 |
| 180° | π | 1/2 |
| 360° | 2π | 1 full circle |
Arc Length Examples
Example 1: Find arc length from radius and angle
You have a circle with a radius of 10 cm and a central angle of 60°.
Convert 60° to radians:
60^\circ = \frac{\pi}{3} \approx 1.0472Apply the formula:
L = 10 * 1.0472 = 10.472
The arc length is approximately 10.472 cm.
Example 2: Find the central angle
You have an arc length of 12 m and a radius of 8 m.
\theta = \frac{12}{8} = 1.5The central angle is 1.5 radians. In degrees:
1.5 * \frac{180}{\pi} \approx 85.9437^\circThe central angle is approximately 85.9437°.
Arc Length FAQ
Do I need to use radians or degrees for arc length?
The standard formula L = rθ requires the angle to be in radians. If your angle is in degrees, convert it to radians first using radians = degrees × π / 180. The calculator lets you enter either degrees or radians and handles the conversion.
Can arc length be larger than the radius?
Yes. Arc length depends on both radius and central angle. For example, if the radius is 5 and the angle is 3 radians, the arc length is 15. Larger angles create longer arcs.
What is the difference between arc length and chord length?
Arc length is the curved distance along the circle. Chord length is the straight-line distance between the two endpoints of the arc. Except for very small angles, the arc length is longer than the chord length.
