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Circle Sector and Segment Calculator

Computes, for a sector of a circle defined by its radius and central angle, the arc length, chord length, sector area, circular segment area and height, and the perimeters of the sector and segment.

When to use

You know a circle's radius and a central angle (degrees or radians) and need the arc, chord, sector or segment measurements.

Do not use when: You need whole-circle properties from radius, diameter or circumference (use circle), or a regular polygon inscribed in a circle (use regular-polygon).

Formula

θ = angle × π/180 (degrees); arc = r·θ; chord = 2r·sin(θ/2); sector_area = r²·θ/2; segment_area = r²·(θ − sin θ)/2; sagitta = r·(1 − cos(θ/2)); sector_perimeter = 2r + arc; segment_perimeter = arc + chord

Inputs

Parameter	Type	Required	Description
`radius`	number	yes	Radius of the circle (any length unit; areas are in that unit squared). Range: > 0, ≤ 1000000000
`angle`	number	yes	Central angle of the sector, at most 360° (2π rad). Range: > 0
`unit`	enum: degrees \| radians	default degrees	Unit of the central angle.

Outputs

Output	Type	Unit	Description
`arc_length`	number		r·θ (θ in radians).
`chord_length`	number		2·r·sin(θ/2): straight line between the arc's end points.
`sector_area`	number		r²·θ/2 (the 'pizza slice').
`segment_area`	number		r²·(θ − sin θ)/2: area between the chord and the arc.
`segment_height`	number		r·(1 − cos(θ/2)): distance from the chord to the arc.
`sector_perimeter`	number		2·r + arc length.
`segment_perimeter`	number		arc length + chord length.
`angle_degrees`	number	°	The central angle in degrees.
`angle_radians`	number	rad	The central angle in radians.
`fraction_of_circle`	number		θ / 2π: the share of the full circle covered by the sector.

Example

r = 10, 90°: {"radius":10,"angle":90,"unit":"degrees"} → {"arc_length":15.707963,"chord_length":14.142136,"sector_area":78.539816,"segment_area":28.539816,"segment_height":2.928932,"sector_perimeter":35.707963,"segment_perimeter":29.850099,"angle_radians":1.570796,"fraction_of_circle":0.25}

r = 6, 60°: {"radius":6,"angle":60} → {"arc_length":6.283185,"chord_length":6,"sector_area":18.849556,"segment_area":3.2611,"segment_height":0.803848,"fraction_of_circle":0.166667}

GET https://tttkmbb.com/api/v1/calculate/circle-sector?radius=10&angle=90&unit=degrees

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/circle-sector (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/circle-sector · Markdown: https://tttkmbb.com/math/circle-sector.md · JSON definition: https://tttkmbb.com/math/circle-sector.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="circle-sector"
OpenAPI operationId: calculate_circle_sector
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Circular sector (reference)
Wikipedia – Circular segment (reference)
Wolfram MathWorld – Circular Sector (reference)

FAQ

What is the difference between a sector and a segment?

A sector is bounded by two radii and the arc (like a pizza slice); a segment is the part of the sector beyond the chord, i.e. sector minus the isosceles triangle formed by the two radii.

Can the angle exceed 180°?

Yes, up to 360°. For angles above 180° the segment is the major segment (larger than a half disc) and the formulas still hold because sin θ becomes negative.

Related calculators

Circle Calculator — Area and circumference of the whole circle.
Angle Converter — Convert the central angle between degrees and radians.
Regular Polygon Calculator — Polygons inscribed in or circumscribed about a circle.