{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"pythagorean"},"result":{"entity_type":"calculator","id":"pythagorean","calculator_id":"pythagorean","canonical_url":"https://tttkmbb.com/math/pythagorean","name":"Pythagorean Theorem Calculator","title":"Pythagorean Theorem Calculator – Find the Missing Side of a Right Triangle (a² + b² = c²)","category":"math","category_name":"Math","tool_name":"solve_pythagorean_theorem","featured_mcp_tool":false,"description":"Finds the missing side of a right triangle from any two of the legs a, b and hypotenuse c using a² + b² = c², and reports the area, perimeter and acute angles.","use_when":"You know two sides of a right triangle and need the third, or want to check whether three lengths form a right triangle.","do_not_use_when":"The triangle is not right-angled (use triangle-solver) or you need the distance between coordinates (use distance-2d).","inputs":[{"name":"a","label":"Leg a","type":"number","required":false,"exclusive_min":0,"description":"First leg (a side adjacent to the right angle). Give any two of a, b, c.","example":3},{"name":"b","label":"Leg b","type":"number","required":false,"exclusive_min":0,"description":"Second leg.","example":4},{"name":"c","label":"Hypotenuse c","type":"number","required":false,"exclusive_min":0,"description":"Hypotenuse (the longest side, opposite the right angle).","example":5}],"outputs":[{"name":"side_a","label":"Leg a","type":"number","decimals":6,"description":"Length of leg a."},{"name":"side_b","label":"Leg b","type":"number","decimals":6,"description":"Length of leg b."},{"name":"hypotenuse_c","label":"Hypotenuse c","type":"number","decimals":6,"description":"Length of the hypotenuse."},{"name":"solved_for","label":"Solved for","type":"string","decimals":4,"description":"Which side was computed ('a', 'b' or 'c'), or a note that all three were given and consistent."},{"name":"area","label":"Area","type":"number","decimals":6,"description":"a × b / 2."},{"name":"perimeter","label":"Perimeter","type":"number","decimals":6,"description":"a + b + c."},{"name":"angle_a_degrees","label":"Angle opposite a","type":"number","unit":"°","decimals":4,"description":"atan(a / b) in degrees."},{"name":"angle_b_degrees","label":"Angle opposite b","type":"number","unit":"°","decimals":4,"description":"atan(b / a) in degrees; the two acute angles sum to 90°."}],"input_schema":{"type":"object","properties":{"a":{"description":"First leg (a side adjacent to the right angle). 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Unit: °.","type":"number","x-unit":"°"},"angle_b_degrees":{"description":"atan(b / a) in degrees; the two acute angles sum to 90°. Unit: °.","type":"number","x-unit":"°"}}},"formula":"c = √(a² + b²); a = √(c² − b²); b = √(c² − a²); area = a·b/2; angle_a = atan(a/b), angle_b = atan(b/a)","sources":[{"name":"Wikipedia – Pythagorean theorem","url":"https://en.wikipedia.org/wiki/Pythagorean_theorem","type":"reference","retrieved_at":"2026-09-23"},{"name":"Wolfram MathWorld – Pythagorean Theorem","url":"https://mathworld.wolfram.com/PythagoreanTheorem.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 = 3, b = 4","inputs":{"a":3,"b":4},"expected":{"side_a":3,"side_b":4,"hypotenuse_c":5,"solved_for":"c","area":6,"perimeter":12,"angle_a_degrees":36.8699,"angle_b_degrees":53.1301},"url":"https://tttkmbb.com/api/v1/calculate/pythagorean?a=3&b=4"},{"name":"a = 5, c = 13","inputs":{"a":5,"c":13},"expected":{"side_b":12,"solved_for":"b","area":30,"perimeter":30,"angle_a_degrees":22.6199},"url":"https://tttkmbb.com/api/v1/calculate/pythagorean?a=5&c=13"}],"faq":[{"q":"What if I enter all three sides?","a":"They are checked against a² + b² = c². Consistent values are returned unchanged; otherwise an error reports the hypotenuse the two legs imply."},{"q":"Why must the hypotenuse be the longest side?","a":"Since c² = a² + b², c is always longer than either leg. A leg equal to or longer than c gives a negative square and no real triangle."}],"tags":["pythagorean theorem","hypotenuse","right triangle","a2 + b2 = c2","missing side"],"related":[{"calculator_id":"right-triangle","reason":"Solve a right triangle from sides and angles."},{"calculator_id":"distance-2d","reason":"Distance between points uses the same theorem."},{"calculator_id":"triangle-solver","reason":"General triangles (law of sines and cosines)."}],"links":{"html":"https://tttkmbb.com/math/pythagorean","markdown":"https://tttkmbb.com/math/pythagorean.md","json":"https://tttkmbb.com/math/pythagorean.json","api":"https://tttkmbb.com/api/v1/calculate/pythagorean","schema":"https://tttkmbb.com/api/v1/calculators/pythagorean","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-23"},"timestamp":"2026-09-23T23:23:14Z"}