{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"circle-sector"},"result":{"entity_type":"calculator","id":"circle-sector","calculator_id":"circle-sector","canonical_url":"https://tttkmbb.com/math/circle-sector","name":"Circle Sector and Segment Calculator","title":"Circle Sector Calculator – Arc Length, Chord, Sector Area, Segment Area and Perimeter from Radius and Angle","category":"math","category_name":"Math","tool_name":"calculate_circle_sector","featured_mcp_tool":false,"description":"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.","use_when":"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).","inputs":[{"name":"radius","label":"Radius","type":"number","required":true,"max":1000000000,"exclusive_min":0,"description":"Radius of the circle (any length unit; areas are in that unit squared).","example":10},{"name":"angle","label":"Central angle","type":"number","required":true,"exclusive_min":0,"description":"Central angle of the sector, at most 360° (2π rad).","example":90},{"name":"unit","label":"Angle unit","type":"enum","required":false,"default":"degrees","values":[{"value":"degrees","label":"Degrees","aliases":["deg","°"]},{"value":"radians","label":"Radians","aliases":["rad"]}],"description":"Unit of the central angle.","example":"degrees"}],"outputs":[{"name":"arc_length","label":"Arc length","type":"number","decimals":6,"description":"r·θ (θ in radians)."},{"name":"chord_length","label":"Chord length","type":"number","decimals":6,"description":"2·r·sin(θ/2): straight line between the arc's end points."},{"name":"sector_area","label":"Sector area","type":"number","decimals":6,"description":"r²·θ/2 (the 'pizza slice')."},{"name":"segment_area","label":"Segment area","type":"number","decimals":6,"description":"r²·(θ − sin θ)/2: area between the chord and the arc."},{"name":"segment_height","label":"Segment height (sagitta)","type":"number","decimals":6,"description":"r·(1 − cos(θ/2)): distance from the chord to the arc."},{"name":"sector_perimeter","label":"Sector perimeter","type":"number","decimals":6,"description":"2·r + arc length."},{"name":"segment_perimeter","label":"Segment perimeter","type":"number","decimals":6,"description":"arc length + chord length."},{"name":"angle_degrees","label":"Central angle","type":"number","unit":"°","decimals":6,"description":"The central angle in degrees."},{"name":"angle_radians","label":"Central angle (radians)","type":"number","unit":"rad","decimals":6,"description":"The central angle in radians."},{"name":"fraction_of_circle","label":"Fraction of the circle","type":"number","decimals":6,"description":"θ / 2π: the share of the full circle covered by the sector."}],"input_schema":{"type":"object","properties":{"radius":{"description":"Radius of the circle (any length unit; areas are in that unit squared).","type":"number","maximum":1000000000,"exclusiveMinimum":0,"examples":[10]},"angle":{"description":"Central angle of the sector, at most 360° (2π rad).","type":"number","exclusiveMinimum":0,"examples":[90]},"unit":{"description":"Unit of the central angle.","type":"string","enum":["degrees","radians"],"default":"degrees","examples":["degrees"]}},"additionalProperties":false,"required":["radius","angle"]},"output_schema":{"type":"object","properties":{"arc_length":{"description":"r·θ (θ in radians).","type":"number"},"chord_length":{"description":"2·r·sin(θ/2): straight line between the arc's end points.","type":"number"},"sector_area":{"description":"r²·θ/2 (the 'pizza slice').","type":"number"},"segment_area":{"description":"r²·(θ − sin θ)/2: area between the chord and the arc.","type":"number"},"segment_height":{"description":"r·(1 − cos(θ/2)): distance from the chord to the arc.","type":"number"},"sector_perimeter":{"description":"2·r + arc length.","type":"number"},"segment_perimeter":{"description":"arc length + chord length.","type":"number"},"angle_degrees":{"description":"The central angle in degrees. Unit: °.","type":"number","x-unit":"°"},"angle_radians":{"description":"The central angle in radians. Unit: rad.","type":"number","x-unit":"rad"},"fraction_of_circle":{"description":"θ / 2π: the share of the full circle covered by the sector.","type":"number"}}},"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","sources":[{"name":"Wikipedia – Circular sector","url":"https://en.wikipedia.org/wiki/Circular_sector","type":"reference","retrieved_at":"2026-09-24"},{"name":"Wikipedia – Circular segment","url":"https://en.wikipedia.org/wiki/Circular_segment","type":"reference","retrieved_at":"2026-09-24"},{"name":"Wolfram MathWorld – Circular Sector","url":"https://mathworld.wolfram.com/CircularSector.html","type":"reference","retrieved_at":"2026-09-24"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output."},"examples":[{"name":"r = 10, 90°","inputs":{"radius":10,"angle":90,"unit":"degrees"},"expected":{"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},"url":"https://tttkmbb.com/api/v1/calculate/circle-sector?radius=10&angle=90&unit=degrees"},{"name":"r = 6, 60°","inputs":{"radius":6,"angle":60},"expected":{"arc_length":6.283185,"chord_length":6,"sector_area":18.849556,"segment_area":3.2611,"segment_height":0.803848,"fraction_of_circle":0.166667},"url":"https://tttkmbb.com/api/v1/calculate/circle-sector?radius=6&angle=60"}],"faq":[{"q":"What is the difference between a sector and a segment?","a":"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."},{"q":"Can the angle exceed 180°?","a":"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."}],"tags":["circle sector","arc length","chord length","sector area","circular segment","sagitta"],"related":[{"calculator_id":"circle","reason":"Area and circumference of the whole circle."},{"calculator_id":"angle","reason":"Convert the central angle between degrees and radians."},{"calculator_id":"regular-polygon","reason":"Polygons inscribed in or circumscribed about a circle."}],"links":{"html":"https://tttkmbb.com/math/circle-sector","markdown":"https://tttkmbb.com/math/circle-sector.md","json":"https://tttkmbb.com/math/circle-sector.json","api":"https://tttkmbb.com/api/v1/calculate/circle-sector","schema":"https://tttkmbb.com/api/v1/calculators/circle-sector","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T01:52:03Z","next_actions":[{"tool":"run_calculator","calculator_id":"circle-sector","reason":"Run Circle Sector and Segment Calculator with the inputs above."}],"links":{"markdown":"https://tttkmbb.com/math/circle-sector.md"}}