Central Angle Calculator

Reviewed by Bojan Žugec, Mathematics (FOI) • Last updated 2026-06-30

The Central Angle Calculator determines the central angle in degrees or radians using arc length, radius, or sector area.

Central Angle Calculator Compute the central angle of a circle using arc length and radius, or by chord length. Choose your known values, enter them below, then select Calculate.

Example Presets

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What Is a Central Angle Calculator?

A central angle is an angle whose vertex is at the center of a circle and whose sides (rays) intercept an arc on the circle. It measures how “wide” that arc is as seen from the center. The angle can be expressed in radians or degrees.

This calculator finds the central angle from the measurements you have. It accepts radius and arc length, radius and chord length, or radius and sector area. It can also work from the circle’s circumference if that is more convenient. The tool converts units, handles radians or degrees, and shows the key steps and formula used.

You might use it for geometry homework, drafting circular parts, laying out arcs in construction, or checking navigation bearings on circular paths. The goal is quick results with clear definitions and reliable methods.

Central Angle Calculator — Run the numbers on central angle.

Formulas for Central Angle

The central angle depends on how much of the circle’s boundary the arc covers, and the radius of the circle. Different input pairs lead to different but equivalent formulas. Here are the most common ones:

From arc length s and radius r (radians): θ = s / r.
From arc length s and radius r (degrees): θ = (s / r) × 180/π.
From chord length c and radius r (radians): θ = 2 × arcsin(c / (2r)).
From sector area A and radius r (radians): θ = 2A / r².
From arc length s and circumference C (degrees): θ = 360° × s / C = 360° × s / (2πr).
From fraction f of a full circle: θ = 2πf (radians) or θ = 360° × f (degrees).

Angles in radians are directly proportional to arc length, which is why θ = s / r is so compact. If you use the chord-based formula, ensure the ratio c / (2r) lies between 0 and 1 to make arcsin valid. When switching between radians and degrees, multiply or divide by 180/π as needed.

How the Central Angle Method Works

Each formula comes from a geometric relationship. The idea is the same: compare your part of the circle to the whole circle. Radians measure arc length relative to radius, while degrees divide the circle into 360 equal parts.

Arc-based approach: A full circle has arc length 2πr. Your arc s is some fraction of that, so θ (radians) = s / r.
Chord-based approach: The chord spans the arc endpoints. In the isosceles triangle formed by the two radii and the chord, c = 2r × sin(θ/2). Solve for θ.
Area-based approach: A sector is a “slice” of the circle, and A = ½r²θ in radians. Rearranging gives θ = 2A / r².
Circumference-based approach: If only C is known, combine s with C to find the fractional arc and then scale by 360° for degrees.
Unit handling: Degrees relate to radians by 180° = π radians. Convert once you obtain an initial result.

These methods give consistent results when inputs are exact. Numerical rounding and measurement error cause small differences, which the calculator reports with your chosen precision.

Inputs, Assumptions & Parameters

The calculator needs one circle size measure and one arc-related measure. Pick the pair you actually have from your problem or design. The tool also needs to know the output units and any precision options.

Radius r: The distance from the circle’s center to the boundary.
Arc length s: The length along the circle between the two radii that define the angle.
Chord length c: The straight-line distance between the arc endpoints.
Sector area A: The area of the slice of the circle bounded by the radii and the arc.
Circumference C or diameter d: An alternative way to provide the circle’s size (C = 2πr; d = 2r).
Angle units and precision: Choose radians or degrees, and set rounding preferences.

Ranges and edge cases matter. All length and area values must be nonnegative, and r must be positive. For chord calculations, c must be ≤ 2r. If s exceeds the circumference, the angle exceeds a full turn; the calculator can return angles beyond 2π radians (or 360°) if desired, or the principal angle if you choose that option.

How to Use the Central Angle Calculator (Steps)

Here’s a concise overview before we dive into the key points:

Choose the input pair you know: arc, chord, or sector area, plus the radius or equivalent.
Enter the radius r, or enter diameter d or circumference C if that is what you have.
Enter the arc length s, chord length c, or sector area A as appropriate.
Select your preferred output units for the angle: radians or degrees.
Set your precision and rounding options if available.
Click Calculate to compute the central angle using the correct formula.

These points provide quick orientation—use them alongside the full explanations in this page.

Worked Examples

A bike wheel of radius r = 0.30 m has an arc marked on its rim with length s = 0.42 m. Using θ = s / r, the angle in radians is θ = 0.42 / 0.30 = 1.4 rad. Converting to degrees gives θ = 1.4 × 180/π ≈ 80.21°. The calculator reports 1.4 rad and 80.21° with your chosen rounding. What this means: the marked arc spans about 80° of the wheel.

A circular window has radius r = 0.50 m. The straight bar across a portion of the window measures c = 0.60 m. Using θ = 2 × arcsin(c / (2r)), θ = 2 × arcsin(0.60 / 1.00) = 2 × arcsin(0.60) ≈ 2 × 0.6435 = 1.287 rad. In degrees, θ ≈ 1.287 × 180/π ≈ 73.77°. What this means: the bar subtends roughly 74° at the window’s center.

Accuracy & Limitations

Results depend on input quality, consistent units, and the chosen formula. In practice, measurement noise and rounding can shift the angle slightly. The differences are usually small, but be aware of your tolerance needs.

Measurement error: Small errors in r or s propagate directly into θ. Calibrate tools when precision is critical.
Chord sensitivity: The arcsin formula magnifies errors when c is close to 0 or 2r.
Unit mismatches: Mixing centimeters and meters, or degrees and radians, causes incorrect outputs.
Large arcs: If s is near or above the full circumference, decide whether you want a principal angle or a multi-turn value.

For high-precision engineering, measure lengths carefully and consider using radians, which link angle and arc length directly. When inputs come from drawings, confirm the scale and rounding conventions.

Units Reference

Consistent units keep calculations correct. A mismatch between radius units and arc units, or confusion between radians and degrees, is a frequent source of errors. The table below lists common quantities and units used in central-angle work.

Common quantities and units for central-angle calculations
Quantity	Symbol	SI unit	Common alternatives	Notes
Central angle	θ	rad	deg	Convert using 180° = π rad.
Arc length	s	m	cm, mm, in	Units must match those used for r.
Radius	r	m	cm, mm, in	Use the same length unit as for s, c, and A.
Chord length	c	m	cm, mm, in	Must satisfy c ≤ 2r.
Sector area	A	m²	cm², mm², in²	In radians, A = ½r²θ.

Read the table by matching your quantity and symbol to its unit. If you use centimeters for r, use centimeters for s and c, and square centimeters for A. Convert to a single unit system before computing.

Common Issues & Fixes

Most problems come from unit mix-ups, invalid chord inputs, or confusing degrees with radians. Check these first before re-entering values.

Angle off by a factor of 57.3: You probably used radians in a degrees-only step (or the reverse).
Math error for arcsin: Ensure c / (2r) is between 0 and 1, and that r > 0.
Strange large angle: Confirm that s is not larger than the circumference unless multi-turn angles are intended.
Inconsistent units: Convert inches to meters or centimeters as needed so r and s match.

If results still look wrong, re-check the formula choice based on your inputs. The arc-length method is usually the simplest when s and r are known.

FAQ about Central Angle Calculator

What inputs give the most stable results?

Arc length and radius are best because θ = s / r is direct and avoids inverse trigonometric steps that can amplify noise.

Can the calculator handle angles larger than a full circle?

Yes. If arc length exceeds the circumference, the computed angle can exceed 2π radians (or 360°). You can also choose the principal angle if preferred.

Which is better for formulas, radians or degrees?

Radians are better for formulas because arc length and area relations become simple. Convert to degrees for reporting if needed.

How accurate are results from chord measurements?

Accuracy is good for mid-range angles. For very small or very large angles, small errors in c cause larger relative errors in θ.

Glossary for Central Angle

Central angle

An angle with vertex at a circle’s center formed by two radii; it subtends an arc on the circle.

Intercepted arc

The portion of the circle bounded by the two rays of the central angle; its length relates to the angle.

Chord

A straight segment connecting two points on a circle; for angle θ and radius r, c = 2r × sin(θ/2).

Radius

The distance from the center to any point on the circle; it sets the circle’s size and scales arc length.

Diameter

A chord passing through the center; equal to 2r. It is the largest chord in the circle.

Sector

A “slice” of a circle bounded by two radii and the intercepted arc; area A = ½r²θ (radians).

Radians

An angle unit defined so that θ (rad) equals arc length divided by radius; 2π radians equals one full circle.

Degrees

An angle unit dividing a circle into 360 parts; 360° equals one full circle and 180° = π radians.

References