Central Angle Calculator
Convert a circular arc length and radius into its subtended central angle.
How the result changes
Only arc length changes; all other inputs stay at your current values. The line illustrates the formula across the shown range. This is not a forecast of real-world outcomes.
About this calculation
Central Angle Calculator: Convert a circular arc length and radius into its subtended central angle. Enter arc length, circle radius; the reported quantities are central angle (degrees), central angle (radians). The equations and examples below show how those inputs lead to the result and where the model's assumptions matter.
Formula
angle in radians = arc length / radius
Worked example
With Arc length: 5 m, Circle radius: 10 m, central angle is 28.64789 degrees.
Method and inputs
An arc of length s on a circle of radius r subtends theta = s/r radians at the center. Multiply radians by 180/pi to get degrees. Arc length and radius must use the same linear unit so their ratio is dimensionless.
Worked example
With a 5-meter arc on a circle of radius 10 meters, theta is 0.5 radian, about 28.65 degrees. Doubling the arc while keeping the radius fixed doubles the angle.
The radius sets the scale of arc length. A 5-meter arc on a 10-meter radius gives half a radian, whereas the same arc on a 20-meter radius gives one quarter radian. This is why arc length alone cannot determine central angle.
How to read the result
An angle above 360 degrees means the stated arc wraps around the circumference more than once. That can be meaningful for total travel along a circular path, but a simple geometric sector is usually one revolution or less.
To check a sector that should be one quarter of a circle, its arc length should be one quarter of the circumference, pi times radius divided by two. Entering that arc produces pi/2 radians or 90 degrees. A larger answer can indicate a chord was measured as an arc.
Assumptions and checks
The formula assumes a circular arc with constant radius. It is not valid for an elliptical curve or arbitrary bend. The radius must be positive; a zero-length arc correctly gives a zero angle.
Do not use chord length in place of arc length. A straight chord is shorter than the curved arc except at zero angle, and requires a different inverse-trigonometric equation. Mixing centimeters and meters also changes the result by a factor of one hundred.
Common questions
What does the central angle calculator calculate?
It converts the ratio of arc length to radius into radians and degrees. The radian value is the direct ratio; the degree value is a unit conversion.
When should I recalculate or check another method?
Recalculate if the radius or arc measurement comes from a different circle. For an angle derived from a chord, measure the chord and use the appropriate chord-angle relation instead.
Inputs
- Arc length (m)
- Circle radius (m)