Central Angle Calculator – Arc Length, Radius, and Angle
A central angle is the angle formed at the center of a circle by two radii drawn to the endpoints of an arc. It directly governs what fraction of the full circle the arc spans: a central angle of 90° corresponds to one quarter of the circle, while 180° defines a semicircle. Understanding central angles is foundational in geometry, trigonometry, engineering, and navigation.
The Core Relationship: s = r × θ
The three quantities — arc length (s), radius (r), and central angle (θ) — are linked by one elegant formula:
s = r × θ
where θ is expressed in radians. To work in degrees, convert first: θ_rad = θ_deg × π / 180. This calculator handles the conversion automatically regardless of the unit you choose.
Rearranging gives the other two forms:
- Central angle from arc and radius:
θ = s / r - Arc length from radius and angle:
s = r × θ - Radius from arc and angle:
r = s / θ
Three Solver Modes
Select the mode matching your available data:
- Find Central Angle — enter radius and arc length; the calculator returns the angle in both degrees and radians. Example: r = 5 m, s = 7.854 m gives θ = 1.5708 rad = 90.0°.
- Find Arc Length — enter radius and central angle; the calculator returns the arc length. Example: r = 10 m, θ = 45° gives s = 7.854 m.
- Find Radius — enter arc length and central angle; the calculator returns the radius. Example: s = 15.708 m, θ = 180° gives r = 5 m.
Secondary Outputs Explained
Beyond the primary solved quantity, the calculator computes four additional values that arise naturally from the same geometry:
- Chord Length (c) — the straight-line distance between the two arc endpoints:
c = 2r × sin(θ / 2). The chord is always shorter than the arc length for any angle less than 360°. - Inscribed Angle — by the inscribed angle theorem, any angle inscribed in the circle that subtends the same arc is exactly half the central angle:
θ_inscribed = θ_central / 2. This is one of the most useful results in circle geometry. - Sector Area — the area of the pie-slice region bounded by the two radii and the arc:
A = ½ r² θ(θ in radians). For a full circle this reduces to πr². - Segment Area — the area of the region between the chord and the arc only (sector minus triangle):
A_seg = ½ r² (θ − sin θ). This is smaller than the sector area for any angle below 180°.
Unit Support
Length inputs and outputs support mm, cm, m, km, in, ft, yd, and mi. All values are internally converted to metres for computation and converted back to the selected unit for display. Angles can be entered in degrees (0° to 360°) or radians (0 to 2π). Precision is adjustable from 0 to 10 decimal places to suit both rough estimates and high-accuracy engineering work.
Real-World Applications
Central angle calculations arise in many practical contexts:
- Road and railway design — horizontal curves on roads and rail lines are defined by their radius and central angle; the arc length gives the actual travel distance through the curve.
- Pipe and duct bending — when fabricating bent pipes or ducts, the central angle determines the required bend geometry and the arc length sets the cut length of the material.
- Circular track layout — athletes and engineers use the central angle formula to mark out lanes, timing points, and sector boundaries on circular tracks.
- Gear and wheel design — tooth spacing and pitch arc lengths on circular gears are computed using central angles and radii.
- Navigation and surveying — bearing changes and arc distances on curved survey boundaries are expressed using central angles.
The Interactive Diagram
The live SVG diagram updates as you type, showing the circle with the two radii r drawn to points A and B, the arc s highlighted in blue, the chord c as a dashed line, the central angle θ annotated at the center O, and a shaded sector region. This visual confirms the geometric scenario before you read the numeric results.