Can the central angle exceed 360° here?

The tool accepts repeated circular travel and keeps the total angle, arc length, and turn count. Above 360°, the diagram explicitly shows one revolution as an outline while the results describe all travel.

Is arc length the same number as arc measure?

Arc length uses distance units. Arc measure uses angular units. They coincide numerically only in particular unit choices, such as radians on a unit-radius circle.

Central Angle Calculator

The central angle calculator connects angular travel around a circle with the radius and the distance along its circumference. Supply arc length and radius to find the angle, or switch modes to calculate arc length or radius. Results include degrees, radians, and the fraction and percentage of a full turn. A central angle has its vertex at the circle's center; its two sides are radii reaching the ends of the selected arc.

Connect a circle's radius and arc with the central angle calculator

Arc length is a distance along the curved circumference. Arc measure is an angle, usually expressed in degrees, and describes how much of the circle that arc covers. They are related but cannot be substituted for one another. The same ninety-degree arc is longer on a larger circle. A radian makes that connection explicit: an angle of one radian subtends an arc whose length equals the radius. This page uses radians internally because they give the direct distance formula.

Choose the two quantities you already know

Choose Arc length + radius to find a central angle. Enter both distances in the same unit, such as meters with meters. Arc length may be zero, but the radius must be positive. The tool cannot infer units from a bare number.
Choose Angle + radius to calculate arc length. The angle field is explicitly in degrees; the calculator converts it to radians before multiplication. The resulting arc length uses the same unit as the entered radius.
Choose Angle + arc length to recover the radius. A unique positive radius requires a positive angle and positive arc length. Zero angle together with zero length fits every positive radius, so the tool reports an undefined radius instead of inventing one.
Read the degree and radian outputs together with the turn fraction. Values above one full revolution are retained. The circle outline shows one revolution for repeated travel, and its caption explains that convention; the numerical result is never silently normalized.

Arc-length formulas use radians

If s is arc length, r is radius, and θ is an angle in radians, θ = s/r, s = rθ, and r = s/θ. Degree input is converted using θ = degrees × π/180. Radians convert back using degrees = θ × 180/π.
The fraction of a full turn is degrees/360, equivalently θ/(2π). Multiplying this fraction by 100 gives the percentage of one turn. A result of 150% represents one and a half revolutions; it does not mean the circle's physical area has grown.
For the ordinary single-arc geometry, 0° through 360° spans zero travel through one complete circumference. Larger nonnegative values represent accumulated travel around the same radius. Negative directed rotations are outside this page's unsigned arc-length domain.
Radius is strictly positive. A positive arc with zero angle is impossible for a finite radius, and a positive angle with zero arc would imply a zero radius. These inconsistent inverse inputs receive specific messages, distinct from the indeterminate zero-over-zero case.

Worked examples with distances and turns

An arc length of 10 and radius of 5 give θ = 10/5 = 2 radians, approximately 114.591559°. The arc covers approximately 0.318310 of a turn, or 31.830989%. Rescaling both distances equally leaves the angle unchanged.
A quarter-circle angle of 90° with radius 8 gives θ = π/2 radians and arc length s = 8 × π/2 = 4π, approximately 12.566371 in the shared length unit. The direct expression provides a check independent of the displayed rounded result.
An angle of 60° and arc length 10 imply radius r = 10/(π/3) = 30/π, approximately 9.549297. Dividing by sixty without converting degrees to radians would give a different and incorrect radius.
An angular travel of 540° with radius 2 gives θ = 3π radians and total curved travel 6π, approximately 18.849556. The turn fraction is 1.5 and percentage is 150%. This is accumulated circumference travel, not a single ordinary sector boundary.

Applications and quantities this page does not calculate

Students can connect radian definitions to circle geometry, while an animation designer can translate a planned rotation into travel along a circular path. A wheel exercise may use radius and angular travel to calculate rim distance under an idealized constant-radius model. This page does not infer rolling slip, determine chord length, or calculate sector area. A chord cuts directly between endpoints and is shorter than the corresponding minor arc. An inscribed angle instead places its vertex on the circumference and follows a different theorem.

Units, approximations, and repeated travel

The central angle calculator preserves full internal floating-point precision and rounds ordinary displayed values to six decimal places. Extreme magnitudes use six significant digits, and values close to angular endpoints can retain more digits. Decimal outputs involving π are approximate. Results outside the browser's representable range are rejected rather than shown as infinity or a false zero. The sector drawing communicates the relationship and does not provide a calibrated physical scale. Check that an input distance follows the circumference, and use radius rather than diameter; entering the diameter halves the calculated angle for a fixed arc length.