{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"pyramid"},"result":{"entity_type":"calculator","id":"pyramid","calculator_id":"pyramid","canonical_url":"https://tttkmbb.com/geometry/pyramid","name":"Square Pyramid Calculator","title":"Square Pyramid Calculator – Volume, Slant Height, Lateral and Total Surface Area","category":"geometry","category_name":"Geometry","tool_name":"calculate_square_pyramid_properties","featured_mcp_tool":false,"description":"Computes the volume, slant height, lateral edge, base area, lateral and total surface area and face slope in degrees of a right square pyramid from its base side and height.","use_when":"You need the volume or surface of a pyramid with a square base and the apex above the centre of the base.","do_not_use_when":"The base is a circle (use cone), a non-square rectangle or triangle, or the apex is off-centre (oblique pyramid).","inputs":[{"name":"base_side","label":"Base side","type":"number","unit":"units","required":true,"exclusive_min":0,"description":"Length of one side of the square base.","example":6},{"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":"base_side² × height / 3."},{"name":"slant_height","label":"Slant height","type":"number","unit":"units","decimals":4,"description":"√(height² + (base_side / 2)²), height of each triangular face."},{"name":"lateral_edge","label":"Lateral edge","type":"number","unit":"units","decimals":4,"description":"√(height² + base_side² / 2), length of each edge from a base corner to the apex."},{"name":"base_area","label":"Base area","type":"number","unit":"units²","decimals":4,"description":"base_side²."},{"name":"lateral_area","label":"Lateral surface area","type":"number","unit":"units²","decimals":4,"description":"2 × base_side × slant_height (four triangular faces)."},{"name":"total_surface_area","label":"Total surface area","type":"number","unit":"units²","decimals":4,"description":"base_side² + 2 × base_side × slant_height."},{"name":"face_slope_deg","label":"Face slope","type":"number","unit":"degrees","decimals":4,"description":"Angle between a triangular face and the base = atan(height / (base_side / 2))."}],"input_schema":{"type":"object","properties":{"base_side":{"description":"Length of one side of the square base. Unit: units.","type":"number","exclusiveMinimum":0,"examples":[6],"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":["base_side","height"]},"output_schema":{"type":"object","properties":{"volume":{"description":"base_side² × height / 3. Unit: units³.","type":"number","x-unit":"units³"},"slant_height":{"description":"√(height² + (base_side / 2)²), height of each triangular face. Unit: units.","type":"number","x-unit":"units"},"lateral_edge":{"description":"√(height² + base_side² / 2), length of each edge from a base corner to the apex. Unit: units.","type":"number","x-unit":"units"},"base_area":{"description":"base_side². Unit: units².","type":"number","x-unit":"units²"},"lateral_area":{"description":"2 × base_side × slant_height (four triangular faces). Unit: units².","type":"number","x-unit":"units²"},"total_surface_area":{"description":"base_side² + 2 × base_side × slant_height. Unit: units².","type":"number","x-unit":"units²"},"face_slope_deg":{"description":"Angle between a triangular face and the base = atan(height / (base_side / 2)). Unit: degrees.","type":"number","x-unit":"degrees"}}},"formula":"volume = base_side² · height / 3; slant_height = √(height² + (base_side/2)²); lateral_edge = √(height² + base_side²/2); lateral_area = 2 · base_side · slant_height; total_surface_area = base_side² + lateral_area; face_slope_deg = atan(2 · height / base_side)","sources":[{"name":"Wikipedia – Square pyramid","url":"https://en.wikipedia.org/wiki/Square_pyramid","type":"reference","retrieved_at":"2026-09-23"},{"name":"Wolfram MathWorld – Square Pyramid","url":"https://mathworld.wolfram.com/SquarePyramid.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":"Base 6, height 4","inputs":{"base_side":6,"height":4},"expected":{"volume":48,"slant_height":5,"lateral_edge":5.831,"base_area":36,"lateral_area":60,"total_surface_area":96,"face_slope_deg":53.1301},"url":"https://tttkmbb.com/api/v1/calculate/pyramid?base_side=6&height=4"},{"name":"Great Pyramid of Giza (230.4 m base, 146.6 m original height)","inputs":{"base_side":230.4,"height":146.6},"expected":{"volume":2594046,"slant_height":186.4473,"lateral_area":85914.92,"face_slope_deg":51.8393},"url":"https://tttkmbb.com/api/v1/calculate/pyramid?base_side=230.4&height=146.6"}],"faq":[{"q":"Slant height or lateral edge?","a":"The slant height runs from the apex to the midpoint of a base side (the height of a face); the lateral edge runs from the apex to a base corner and is longer."},{"q":"What about a rectangular base?","a":"Volume is length × width × height / 3, but the two pairs of faces have different slant heights; this calculator only handles square bases."}],"tags":["pyramid","square pyramid","volume of a pyramid","slant height","surface area of a pyramid"],"related":[{"calculator_id":"cone","reason":"Circular-based counterpart with the same volume formula (base area × height / 3)."},{"calculator_id":"rectangular-prism","reason":"A prism with the same base and height has three times the volume."},{"calculator_id":"right-triangle","reason":"Slant height and lateral edge come from right triangles with the height as one leg."}],"links":{"html":"https://tttkmbb.com/geometry/pyramid","markdown":"https://tttkmbb.com/geometry/pyramid.md","json":"https://tttkmbb.com/geometry/pyramid.json","api":"https://tttkmbb.com/api/v1/calculate/pyramid","schema":"https://tttkmbb.com/api/v1/calculators/pyramid","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-23"},"timestamp":"2026-09-23T23:25:41Z","next_actions":[{"tool":"run_calculator","calculator_id":"pyramid","reason":"Run Square Pyramid Calculator with the inputs above."}],"links":{"markdown":"https://tttkmbb.com/geometry/pyramid.md"}}