{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"cone"},"result":{"entity_type":"calculator","id":"cone","calculator_id":"cone","canonical_url":"https://tttkmbb.com/geometry/cone","name":"Cone Calculator","title":"Cone Calculator – Volume, Slant Height, Lateral and Total Surface Area from Radius and Height","category":"geometry","category_name":"Geometry","tool_name":"calculate_cone_properties","featured_mcp_tool":false,"description":"Computes the volume, slant height, base area, lateral (curved) surface area, total surface area and half-apex angle in degrees of a right circular cone from its base radius and height.","use_when":"You need the capacity or surface of a funnel, cone-shaped pile, roof or any right circular cone.","do_not_use_when":"The cone is truncated (frustum), oblique, or has a square base (use pyramid).","inputs":[{"name":"radius","label":"Radius","type":"number","unit":"units","required":true,"exclusive_min":0,"description":"Radius of the circular base.","example":3},{"name":"height","label":"Height","type":"number","unit":"units","required":true,"exclusive_min":0,"description":"Perpendicular distance from the base to the apex.","example":4}],"outputs":[{"name":"volume","label":"Volume","type":"number","unit":"units³","decimals":4,"description":"1/3 × π × r² × h."},{"name":"slant_height","label":"Slant height","type":"number","unit":"units","decimals":4,"description":"√(r² + h²), distance from the apex to the edge of the base."},{"name":"base_area","label":"Base area","type":"number","unit":"units²","decimals":4,"description":"π × r²."},{"name":"lateral_area","label":"Lateral surface area","type":"number","unit":"units²","decimals":4,"description":"π × r × slant_height, the curved surface only."},{"name":"total_surface_area","label":"Total surface area","type":"number","unit":"units²","decimals":4,"description":"π × r × (r + slant_height)."},{"name":"half_apex_angle_deg","label":"Half apex angle","type":"number","unit":"degrees","decimals":4,"description":"Angle between the axis and the slant side = atan(r / h)."}],"input_schema":{"type":"object","properties":{"radius":{"description":"Radius of the circular base. Unit: units.","type":"number","exclusiveMinimum":0,"examples":[3],"x-unit":"units"},"height":{"description":"Perpendicular distance from the base to the apex. Unit: units.","type":"number","exclusiveMinimum":0,"examples":[4],"x-unit":"units"}},"additionalProperties":false,"required":["radius","height"]},"output_schema":{"type":"object","properties":{"volume":{"description":"1/3 × π × r² × h. Unit: units³.","type":"number","x-unit":"units³"},"slant_height":{"description":"√(r² + h²), distance from the apex to the edge of the base. Unit: units.","type":"number","x-unit":"units"},"base_area":{"description":"π × r². Unit: units².","type":"number","x-unit":"units²"},"lateral_area":{"description":"π × r × slant_height, the curved surface only. Unit: units².","type":"number","x-unit":"units²"},"total_surface_area":{"description":"π × r × (r + slant_height). Unit: units².","type":"number","x-unit":"units²"},"half_apex_angle_deg":{"description":"Angle between the axis and the slant side = atan(r / h). Unit: degrees.","type":"number","x-unit":"degrees"}}},"formula":"volume = π · radius² · height / 3; slant_height = √(radius² + height²); base_area = π · radius²; lateral_area = π · radius · slant_height; total_surface_area = π · radius · (radius + slant_height); half_apex_angle_deg = atan(radius / height)","sources":[{"name":"Wikipedia – Cone","url":"https://en.wikipedia.org/wiki/Cone","type":"reference","retrieved_at":"2026-09-23"},{"name":"Wolfram MathWorld – Cone","url":"https://mathworld.wolfram.com/Cone.html","type":"reference","retrieved_at":"2026-09-23"}],"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":"Radius 3, height 4","inputs":{"radius":3,"height":4},"expected":{"volume":37.6991,"slant_height":5,"base_area":28.2743,"lateral_area":47.1239,"total_surface_area":75.3982,"half_apex_angle_deg":36.8699},"url":"https://tttkmbb.com/api/v1/calculate/cone?radius=3&height=4"},{"name":"Radius 5, height 12","inputs":{"radius":5,"height":12},"expected":{"volume":314.1593,"slant_height":13,"lateral_area":204.2035,"total_surface_area":282.7433,"half_apex_angle_deg":22.6199},"url":"https://tttkmbb.com/api/v1/calculate/cone?radius=5&height=12"}],"faq":[{"q":"Height or slant height?","a":"Enter the perpendicular height (base to apex along the axis). If you only know the slant height l, the height is √(l² − r²)."},{"q":"Why is the volume a third of a cylinder's?","a":"A cone with the same base and height has exactly one third of the cylinder's volume; this follows from integrating the shrinking cross-sections."}],"tags":["cone","volume of a cone","slant height","surface area of a cone","lateral area"],"related":[{"calculator_id":"cylinder","reason":"Same base without the taper (three times the volume)."},{"calculator_id":"pyramid","reason":"The square-based counterpart."},{"calculator_id":"right-triangle","reason":"The cross-section through the axis is a right triangle with legs r and h."}],"links":{"html":"https://tttkmbb.com/geometry/cone","markdown":"https://tttkmbb.com/geometry/cone.md","json":"https://tttkmbb.com/geometry/cone.json","api":"https://tttkmbb.com/api/v1/calculate/cone","schema":"https://tttkmbb.com/api/v1/calculators/cone","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-23"},"timestamp":"2026-09-23T23:23:07Z"}