About

A central angle θ is the angle subtended at the center of a circle by an arc or chord. Miscalculating it propagates errors into sector area, arc length, and segment geometry - critical in CAD drafting, gear tooth design, and land surveying where angular tolerances below 0.1° matter. This calculator derives θ from three independent input pairs: arc length s with radius r, chord length c with radius r, or sector area A with radius r. All results are cross-validated and rendered as an interactive SVG diagram.

The underlying trigonometry uses the IEEE 754 double-precision arcsin implementation native to the browser. Precision is limited to approximately 15 significant digits. Note: the chord-based formula is undefined when c > 2r (chord cannot exceed diameter). The tool validates this constraint before computation.

Formulas

The central angle θ can be derived from three geometric properties of a circle with radius r:

Method 1 - From Arc Length:

θ = sr rad

Method 2 - From Chord Length:

θ = 2 arcsin(c2r) rad

Method 3 - From Sector Area:

θ = 2Ar² rad

Degree conversion:

θ_deg = θ_rad × 180π

Where s = arc length, r = radius, c = chord length, A = sector area, θ = central angle. The chord method constrains c ≤ 2r (chord cannot exceed diameter). For the arc length method, s ≤ 2πr limits the angle to a full circle. The sector area method constrains A ≤ πr².

Reference Data

Input Method	Required Inputs	Formula	Valid Range for θ	Common Use Case
Arc Length + Radius	s, r	θ = sr	0 - 2π rad	Road curves, track design
Chord Length + Radius	c, r	θ = 2 arcsin(c2r)	0 - π rad	Structural arches, bridges
Sector Area + Radius	A, r	θ = 2Ar²	0 - 2π rad	Pizza slicing, land plots
Common Central Angle Reference Values
Full circle	-	360°	2π rad	Complete rotation
Semicircle	-	180°	π rad	Diameter arc
Quarter circle	-	90°	π2 rad	Right angle sector
Sextant	-	60°	π3 rad	Hexagonal geometry
Octant	-	45°	π4 rad	Compass bearings
30° sector	-	30°	π6 rad	Clock hour marks
1 rad	-	57.2958°	1 rad	Unit radian reference
1°	-	1°	0.01745 rad	Unit degree reference
Gear tooth (20T)	-	18°	π10 rad	Mechanical engineering
Pentagon interior	-	72°	2π5 rad	Regular polygon layout
Protractor half	-	180°	π rad	Standard measurement tool

Frequently Asked Questions

The chord-based formula uses arcsin(c / 2r), which maps to the range [0, π]. A chord of length c subtends two possible central angles: θ and 2π − θ. Without additional information (like arc length), the calculator returns the minor angle. If you need the reflex angle, subtract the result from 360°.

When c = 2r, the chord is the diameter itself. The formula yields arcsin(1) = π/2, so θ = 2 × π/2 = π rad (180°). This is geometrically correct: a diameter always subtends a semicircle.

JavaScript uses IEEE 754 double-precision floats with ~15 significant digits. Near c/2r ≈ 1, the arcsin function's derivative approaches infinity, amplifying rounding errors. For a ratio of 0.99999999, the angular error is approximately 0.0001°. For engineering work requiring sub-arcsecond precision, use dedicated numerical libraries or symbolic computation.

Not directly from a closed-form equation. The relationship s = rθ and c = 2r sin(θ/2) yields the transcendental equation s/c = θ/(2 sin(θ/2)), which requires numerical root-finding (Newton-Raphson). This calculator requires at least the radius as a known input.

Multiply the fractional degree part by 60 to get arcminutes, then multiply the fractional arcminute by 60 for arcseconds. For example, 45.5125° = 45° 30′ 45″. This calculator displays DMS alongside decimal degrees in the results panel.

No. The sector area A is the "pie slice" from the center, bounded by two radii and the arc. The segment area (between the chord and arc) equals A_sector − A_triangle = r²2(θ − sin θ). If you input segment area instead of sector area, the result will be incorrect. The tool computes and displays the segment area as a derived value for cross-checking.