Central Angle Calculator
Enter an arc length and the circle’s radius and get the central angle it subtends — in degrees, radians and gradians — using the exact relationship θ = s ÷ r.
Central angle (degrees)
143.239449°
- Central angle (radians)
- 2.5
- Central angle (gradians)
- 159.154943
- Fraction of full circle
- 39.7887%
- Circumference (2πr)
- 12.566371
The central angle is the angle at the centre of the circle that the arc “opens up” to. In radians it is simply the arc length divided by the radius (θ = s ÷ r) — that ratio is the very definition of the radian. Multiplying by 180/π converts it to degrees. An arc as long as the radius spans exactly one radian (about 57.30°); a full lap of the circle, where the arc equals the circumference 2πr, spans 2π radians = 360°.
How to use this calculator
Type the length of your arc (s) and the radius (r) of the circle it sits on, using any unit you like — centimetres, metres, inches — as long as both use the same one. The calculator returns the central angle the arc subtends at the centre of the circle, shown in degrees, radians and gradians, together with the fraction of the whole circle that the arc covers. Because the angle depends only on the ratio s ÷ r, the unit you choose does not matter as long as it is consistent.
How the calculation works
The central angle is the angle formed at the centre of the circle by the two radii drawn to the ends of the arc. By the definition of the radian, that angle in radians equals the arc length divided by the radius: θ = s ÷ r. To convert to degrees the calculator multiplies by 180/π; for gradians it multiplies by 200/π (a full turn is 400 gon). The fraction of the circle covered is θ ÷ 2π, which is the same as s ÷ (2πr) — the arc length over the circumference. π is used at full machine precision, so rounding happens only when the result is displayed.
Worked example
Suppose an arc is 5 units long on a circle of radius 2 units. Then θ = s ÷ r = 5 ÷ 2 = 2.5 radians. Converting, 2.5 × 180/π ≈ 143.24°. The circumference is 2π·2 ≈ 12.566 units, so the arc covers 5 ÷ 12.566 ≈ 39.79% of the circle. As a sanity check, an arc exactly as long as the radius (s = r) gives θ = 1 radian ≈ 57.30°, and an arc equal to the full circumference (s = 2πr) gives θ = 2π radians = 360°.
Frequently asked questions
What is the formula for a central angle from arc length?
The central angle in radians equals the arc length divided by the radius: θ = s ÷ r. This is the definition of the radian — the angle for which the arc is exactly as long as the radius. To get degrees, multiply the radian value by 180/π (about 57.2958).
What is the difference between a central angle and an inscribed angle?
A central angle has its vertex at the centre of the circle, with its two sides being radii to the ends of the arc. An inscribed angle has its vertex on the circle itself. For the same arc, the inscribed angle theorem says the central angle is exactly twice the inscribed angle. This calculator computes the central angle.
Why do I get the same angle whatever unit I use?
Because the central angle depends only on the ratio of arc length to radius, s ÷ r. If you double both (say switch from centimetres to half-centimetres), the ratio is unchanged, so the angle is unchanged. Just make sure the arc length and the radius are expressed in the same unit within one calculation.
Can the central angle be more than 360 degrees?
For a single sweep around a circle the arc length is at most the circumference (2πr), which gives exactly 360°. If you enter an arc longer than the circumference the formula still returns a value above 360°, which represents more than one full wrap around the circle — mathematically valid, though not a simple sector angle.
How do I convert the answer to radians or gradians?
The calculator shows all three at once. Radians come straight from θ = s ÷ r. Degrees are radians × 180/π. Gradians (also called gon) are radians × 200/π, so a right angle is 100 gon and a full turn is 400 gon. Radians are the natural unit here; degrees are the everyday one.
How is this related to the area of the sector?
Once you know the central angle you can find the sector area with A = ½·r²·θ, where θ is in radians. So an arc of length s on radius r bounds a sector of area ½·r²·(s ÷ r) = ½·r·s. The central angle is the bridge between the arc, the sector and the fraction of the whole circle involved.