{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"ellipse"},"result":{"entity_type":"calculator","id":"ellipse","calculator_id":"ellipse","canonical_url":"https://tttkmbb.com/geometry/ellipse","name":"Ellipse Calculator","title":"Ellipse Calculator – Area, Perimeter (Ramanujan), Eccentricity and Foci from the Semi-Axes","category":"geometry","category_name":"Geometry","tool_name":"calculate_ellipse_properties","featured_mcp_tool":false,"description":"Computes the area, approximate perimeter (Ramanujan's second approximation), eccentricity and focal distances of an ellipse from its semi-major and semi-minor axes.","use_when":"You need the area or circumference of an ellipse or oval, or the position of its foci, from the two half-axes.","do_not_use_when":"Both axes are equal (use circle for exact results) or you need the arc length of part of an ellipse, which requires elliptic integrals.","inputs":[{"name":"semi_major_a","label":"Semi-major axis a","type":"number","unit":"units","required":true,"exclusive_min":0,"description":"Half of the longest diameter (centre to the farthest point).","example":5},{"name":"semi_minor_b","label":"Semi-minor axis b","type":"number","unit":"units","required":true,"exclusive_min":0,"description":"Half of the shortest diameter. If larger than a, the two are swapped.","example":3}],"outputs":[{"name":"area","label":"Area","type":"number","unit":"units²","decimals":4,"description":"π × a × b."},{"name":"perimeter","label":"Perimeter (approx.)","type":"number","unit":"units","decimals":4,"description":"Ramanujan's second approximation; relative error below 4·10⁻⁵ % for a/b ≤ 5 and about 0.04 % as b → 0."},{"name":"eccentricity","label":"Eccentricity","type":"number","decimals":4,"description":"e = √(1 − b² / a²); 0 for a circle, approaching 1 for a very flat ellipse."},{"name":"linear_eccentricity","label":"Centre-to-focus distance","type":"number","unit":"units","decimals":4,"description":"c = √(a² − b²), distance from the centre to each focus."},{"name":"distance_between_foci","label":"Distance between foci","type":"number","unit":"units","decimals":4,"description":"2 × c."}],"input_schema":{"type":"object","properties":{"semi_major_a":{"description":"Half of the longest diameter (centre to the farthest point). Unit: units.","type":"number","exclusiveMinimum":0,"examples":[5],"x-unit":"units"},"semi_minor_b":{"description":"Half of the shortest diameter. If larger than a, the two are swapped. Unit: units.","type":"number","exclusiveMinimum":0,"examples":[3],"x-unit":"units"}},"additionalProperties":false,"required":["semi_major_a","semi_minor_b"]},"output_schema":{"type":"object","properties":{"area":{"description":"π × a × b. Unit: units².","type":"number","x-unit":"units²"},"perimeter":{"description":"Ramanujan's second approximation; relative error below 4·10⁻⁵ % for a/b ≤ 5 and about 0.04 % as b → 0. Unit: units.","type":"number","x-unit":"units"},"eccentricity":{"description":"e = √(1 − b² / a²); 0 for a circle, approaching 1 for a very flat ellipse.","type":"number"},"linear_eccentricity":{"description":"c = √(a² − b²), distance from the centre to each focus. Unit: units.","type":"number","x-unit":"units"},"distance_between_foci":{"description":"2 × c. Unit: units.","type":"number","x-unit":"units"}}},"formula":"area = π·a·b; h = ((a − b) / (a + b))²; perimeter ≈ π·(a + b)·(1 + 3h / (10 + √(4 − 3h))) (Ramanujan 1914, second approximation); eccentricity = √(1 − b²/a²); c = √(a² − b²)","method":"The exact perimeter is a complete elliptic integral of the second kind with no closed form; Ramanujan's second approximation is used because its error is negligible for practical axis ratios.","sources":[{"name":"Wikipedia – Ellipse – circumference","url":"https://en.wikipedia.org/wiki/Ellipse#Circumference","type":"reference","retrieved_at":"2026-09-23"},{"name":"Wolfram MathWorld – Ellipse","url":"https://mathworld.wolfram.com/Ellipse.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":"a = 5, b = 3","inputs":{"semi_major_a":5,"semi_minor_b":3},"expected":{"area":47.1239,"perimeter":25.527,"eccentricity":0.8,"linear_eccentricity":4,"distance_between_foci":8},"url":"https://tttkmbb.com/api/v1/calculate/ellipse?semi_major_a=5&semi_minor_b=3"},{"name":"a = 7, b = 4","inputs":{"semi_major_a":7,"semi_minor_b":4},"expected":{"area":87.9646,"perimeter":35.2032,"eccentricity":0.8207,"linear_eccentricity":5.7446},"url":"https://tttkmbb.com/api/v1/calculate/ellipse?semi_major_a=7&semi_minor_b=4"}],"faq":[{"q":"Do I enter the full axes or the half-axes?","a":"The half-axes (semi-axes), measured from the centre. If you know the full width and height of the oval, halve them first."},{"q":"How accurate is the perimeter?","a":"Ramanujan's second formula is exact for a circle and within 0.00004 % up to an axis ratio of 5:1; the worst case (a degenerate ellipse with b = 0) is about 0.04 % low."}],"tags":["ellipse","area of an ellipse","ellipse perimeter","ramanujan","eccentricity","foci"],"related":[{"calculator_id":"circle","reason":"Exact results when both axes are equal."},{"calculator_id":"area","reason":"Convert the area to another unit."}],"links":{"html":"https://tttkmbb.com/geometry/ellipse","markdown":"https://tttkmbb.com/geometry/ellipse.md","json":"https://tttkmbb.com/geometry/ellipse.json","api":"https://tttkmbb.com/api/v1/calculate/ellipse","schema":"https://tttkmbb.com/api/v1/calculators/ellipse","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-23"},"timestamp":"2026-09-23T23:29:35Z","next_actions":[{"tool":"run_calculator","calculator_id":"ellipse","reason":"Run Ellipse Calculator with the inputs above."}],"links":{"markdown":"https://tttkmbb.com/geometry/ellipse.md"}}