Central Angle Calculator - Arc Length to Angle Solver

A central angle calculator finds the central angle at the center of a circle that sweeps across a given arc. The two values you need are the radius and the curved arc length; the tool then reports the central angle in degrees, radians, or as a fraction of the full 360-degree circle. A central angle calculator like this one is the natural companion to arc length, chord length, and sector area problems where you know the curve but not the angle that produced it.

• • Pie chart planning: Translate a known arc length on a circular chart into the central angle for labels, design software, or geometry homework.
• • Circular construction: Find the central angle that produces a specific arc length on a curved handrail, pipe bend, decorative ring, or stone edge.
• • Navigation and clock-face work: Convert a distance along a circle of given radius into the angle at the center of the dial or compass.
• • Trigonometry homework: Verify textbook answers that ask for the central angle in radians from arc length, especially when the answer needs to feed into other circle formulas.

A central angle is the angle formed at the very center of the circle by two radii that end at the start and end of the arc. The arc length, sector area, and chord length all depend on the same central angle, so once you know it, the rest of the circle geometry falls into place.

The cleanest way to think about the central angle is in radians. When the angle is one radian, the arc length equals the radius. That is what makes the formula theta equals arc length divided by radius so direct, and it lets a small circle and a large circle share the same central angle because the arc length scales with the radius.

If you already know the radius and the central angle and want the curve length, the arc length calculator runs the same circle relationship in the other direction.

The calculator divides the arc length by the radius to get the central angle in radians, then converts to degrees if the answer unit is set to degrees. It also uses the same angle to compute the sector area, the chord length, and the fraction of the circle the arc covers.

• radius: Distance from the center of the circle to the edge, in the shared length unit (cm, m, in, or ft).
• arcLength: Curved distance along the part of the circle, in the same length unit as the radius.
• angleUnit: Whether the primary result is shown in degrees or radians. The formula always computes radians first.

Because the formula relies on a single division, the central angle responds directly to the arc length. Doubling the arc length at the same radius doubles the angle, and halving the radius at the same arc length doubles the angle. That direct proportionality is one reason the radian form of the formula is preferred in class work.

OpenStax Precalculus treats the radian measure of an angle as the arc length divided by the radius, which is the exact operation performed here. The same source confirms that one full rotation of a circle is 2 * pi radians, so a full 360-degree sweep always comes out to that value.

According to OpenStax Precalculus 2e, the radian measure of an angle is the arc length divided by the radius, so theta equals s divided by r when the angle is measured in radians.

To check the straight-line distance across the same arc, the chord length calculator applies the related chord formula and the same radius and angle.

A central angle is the bridge between a curved arc and the rest of the circle's geometry. These four concepts cover the inputs, the formula, and the related outputs.

Radians and degrees measure the same angle, but only radians keep the relationship theta equals arc length divided by radius. That is why a calculator that uses radians internally and only converts at the end is the most reliable way to avoid unit mistakes.

If the arc length is larger than the full circumference, the formula still returns a number, but the geometric interpretation changes. In that case the central angle is a reflex or multi-wrap value, a useful signal that the inputs may not describe a single sweep of the circle.

When the answer needs to switch units outside of this tool, the radians to degrees calculator handles the standalone conversion between radians and degrees.

Enter the radius and the arc length in the same length unit, pick the answer unit, and the calculator returns the central angle plus the related circle measurements.

1. 1 Enter the radius: Type the radius in the first field. Use the same length unit as the arc.
2. 2 Enter the arc length: Type the curved distance. Make sure it is in the same length unit as the radius.
3. 3 Pick the length unit: Choose cm, m, in, or ft so chord, sector area, and circumference share one system.
4. 4 Pick the answer unit: Choose degrees for a protractor or pie chart, or radians if the angle feeds into other circle formulas.
5. 5 Read the result: Use the central angle as the primary answer and review sector area, chord length, and fraction of circle as supporting checks.

A curved handrail needs a 60-degree sweep on a 10 cm radius. The arc length is 10 * pi / 3 = 10.472 cm. Enter radius 10, arc length 10.472, pick centimeters, and the calculator returns theta = 1.0472 rad (60 degrees) with a sector area of 52.36 cm^2 and a chord of 10 cm.

If the input angle is given in a different unit such as gradians or turns, the angle converter normalizes it before you reuse it with the central angle formula.

The main benefit is turning a curved measurement into the angle that the rest of your math needs, without doing the conversion by hand.

• • Direct arc to angle conversion: Skip dividing the arc length by the radius and converting to degrees by hand.
• • Degrees and radians together: See the central angle in both units at once, which is useful when a downstream formula expects radians but your audience reads degrees.
• • Cross-check the geometry: Sector area, chord length, fraction of circle, and full circumference all come from the same radius and angle, so one input set confirms the whole circle.
• • Catch unit mistakes: A fraction of circle above 100 percent, or a chord longer than the diameter, is a fast signal that the radius and arc were entered in different units.
• • Drop into other tools: Drop the central angle directly into spreadsheet formulas, SVG path commands, or chart software.
• • Plan curved work: Size a curved cut, weld, or boundary before ordering material or marking a layout.

For homework and exam prep, the central angle calculator removes the most common slip in circle problems: forgetting to convert between degrees and radians. For applied work, the same conversion is what makes the result line up with downstream circle formulas, so the saved time compounds across longer problem sets.

When the same radius and central angle are part of a larger circle problem, the circle calculator collects the full set of related measurements in one place.

The result depends on the quality of the radius and arc length inputs and on whether the answer unit matches the next step in your problem.

• • Assumes a flat circle. Does not handle spherical triangles or great-circle arcs.
• • The radius must be positive and the arc length must be non-negative; other inputs are rejected before the formula runs.
• • The chord uses 2r * sin(theta / 2). For reflex angles above 180 degrees the formula is still correct, but the visible chord on a single sweep of the circle is shorter than the value because the chord crosses the center.

OpenStax Algebra and Trigonometry ties the radian measure directly to the circle, which is why the same circle formulas work for any positive radius once the angle is in radians. That is why this central angle calculator uses radians internally and only converts at the very end, where the answer unit is selected.

According to OpenStax Algebra and Trigonometry, the radian measure ties an angle directly to the circle, which is why the same circle formulas work for any positive radius once the angle is converted to radians.