Find a circle’s central angle from arc length and radius, or solve for arc length or radius from two known quantities. Radius is half the diameter.
cm To Degrees Formula
The following formula calculates the central angle in degrees from an arc length and circle diameter.
θ (degrees) = 360s/(πD) = (s/r) × 180/π, where D = 2r. Arc length s = rθπ/180. Use matching length units.
Variables:
- theta is the angle in degrees
- d is the arc length in centimeters
- D is the diameter of the circle in centimeters; pi approximately equals 3.14159
Using radius (r = D/2) instead: theta = (s / r) x 57.296 degrees, where 57.296 = 180/pi is the number of degrees in one radian. Both forms are equivalent. The calculator asks for radius: enter half the circle diameter.
| Arc Length (cm) | Angle (deg) |
|---|---|
| 0.5 | 5.730 |
| 1 | 11.459 |
| 2 | 22.918 |
| 3 | 34.377 |
| 4 | 45.837 |
| 5 | 57.296 |
| 6 | 68.755 |
| 7.5 | 85.944 |
| 8 | 91.673 |
| 9 | 103.132 |
| 10 | 114.592 |
| 12 | 137.510 |
| 12.5 | 143.239 |
| 15 | 171.887 |
| 16 | 183.346 |
| 20 | 229.183 |
| 24 | 275.020 |
| 25 | 286.479 |
| 30 | 343.775 |
| 31.416 | 360.001 |
| * Rounded to 3 decimals. Assumes diameter D = 10 cm. At arc length = 5 cm the result is 57.296 deg = 1 radian, confirming the radian relationship. | |
What is cm to Degrees?
Converting centimeters to degrees finds the central angle subtended by an arc of measured length. It is the inverse of the arc length formula (s = r x theta in radians), rearranged as theta = s/r x 57.296 degrees. The rounded factor 57.296 approximates 180/pi, the number of degrees in one radian. The calculator uses Math.PI without rounding the conversion factor. The result is always circle-size-dependent: 1 cm spans 17.1 degrees on a 6.7 cm tennis ball, but only 1.8 degrees on a 63.5 cm car tire. This conversion appears in rotary encoder calibration, CNC arc programming, gear tooth pitch calculation, and any system that maps linear travel along a curved path to a rotation angle.
Arc-to-Angle by Object
How many degrees 1 cm of arc represents on common circular objects:
| Object | Diameter (cm) | Circumference (cm) | 1 cm arc (deg) |
|---|---|---|---|
| Tennis ball | 6.7 | 21.0 | 17.10 |
| Baseball | 7.4 | 23.2 | 15.49 |
| Basketball | 24.1 | 75.7 | 4.75 |
| 12-inch pizza | 30.5 | 95.8 | 3.76 |
| Clock face (30 cm) | 30.0 | 94.2 | 3.82 |
| Car tire (195/65 R15) | 63.5 | 199.5 | 1.80 |
| Bicycle wheel (700c x 23mm) | 66.8 | 209.9 | 1.72 |
| * Formula: degrees per cm = 360 / (pi x D), using the stated diameter. Values rounded to 2 decimal places. These diameters are illustrative approximations; measure the actual object for a precise conversion. | |||
Practical Applications
| Field | How It Is Used | Key Detail |
|---|---|---|
| Rotary encoders | Convert wheel arc travel to shaft rotation angle | Angle resolution = 360 / pulses per revolution; each pulse = fixed arc length on wheel |
| CNC machining | Verify arc toolpath length against G-code sweep angle | G02/G03 arc syntax depends on the controller and uses endpoints with center offsets or radius; arc length = r x theta (radians) |
| Gear design | Calculate circular pitch from tooth count and diameter | Circular pitch (arc per tooth) = pi x D / number of teeth |
| Navigation / geodesy | Angular distance from surface arc measurement | On an assumed sphere of radius 6,371 km, 1 degree of surface arc is about 111.195 km. Actual Earth latitude distance varies. |
| Bicycle computers | Distance from counted wheel rotations | 1 revolution = pi x wheel diameter cm of travel; odometer accuracy depends on exact diameter |
