{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"lens-equation"},"result":{"entity_type":"calculator","id":"lens-equation","calculator_id":"lens-equation","canonical_url":"https://tttkmbb.com/physics/lens-equation","name":"Thin Lens Equation Calculator","title":"Thin Lens Equation Calculator – Image Distance, Magnification and Image Type (Lenses, Mirrors)","category":"physics","category_name":"Physics","tool_name":"solve_thin_lens_equation","featured_mcp_tool":false,"description":"Solves the thin-lens / mirror equation 1/f = 1/d_o + 1/d_i for the image distance of a converging or diverging lens, or a concave or convex mirror, and reports the magnification, image height and whether the image is real or virtual, inverted or upright, enlarged or reduced.","use_when":"You know the focal length of a lens or curved mirror and the object distance and need where the image forms, how large it is and its character (real/virtual, upright/inverted).","do_not_use_when":"The lens is thick or part of a multi-element system, you need the focal length from surface curvatures (lensmaker's equation), or you need refraction angles at a flat surface (use snells-law).","inputs":[{"name":"optic","label":"Optic","type":"enum","required":true,"values":[{"value":"converging_lens","label":"Converging (convex) lens, f > 0","aliases":["convex lens","convex_lens","converging","biconvex"]},{"value":"diverging_lens","label":"Diverging (concave) lens, f < 0","aliases":["concave lens","concave_lens","diverging","biconcave"]},{"value":"concave_mirror","label":"Concave (converging) mirror, f > 0","aliases":["converging mirror"]},{"value":"convex_mirror","label":"Convex (diverging) mirror, f < 0","aliases":["diverging mirror"]}],"description":"Type of optic; it sets the sign of the focal length (converging lens and concave mirror positive, diverging lens and convex mirror negative).","example":"converging_lens"},{"name":"focal_length_cm","label":"Focal length (magnitude)","type":"number","unit":"cm","required":true,"exclusive_min":0,"description":"Magnitude of the focal length in centimetres, always positive; the sign is applied from optic. For a spherical mirror f = R / 2.","example":10},{"name":"object_distance_cm","label":"Object distance","type":"number","unit":"cm","required":true,"exclusive_min":0,"description":"Distance from the object to the lens or mirror in centimetres (a real object, always positive).","example":15},{"name":"object_height_cm","label":"Object height","type":"number","unit":"cm","required":false,"default":1,"exclusive_min":0,"description":"Height of the object in centimetres (default 1, so image_height_cm equals the magnification).","example":1}],"outputs":[{"name":"focal_length_signed_cm","label":"Focal length (signed)","type":"number","unit":"cm","decimals":4,"description":"Focal length with the sign convention applied: positive for converging lens and concave mirror, negative for diverging lens and convex mirror."},{"name":"image_distance_cm","label":"Image distance","type":"number","unit":"cm","decimals":4,"description":"d_i = 1 / (1/f − 1/d_o), signed: positive = real image, negative = virtual image (see image_location_note)."},{"name":"magnification","label":"Magnification","type":"number","decimals":4,"description":"m = −d_i / d_o; negative means inverted, |m| > 1 enlarged."},{"name":"image_height_cm","label":"Image height","type":"number","unit":"cm","decimals":4,"description":"m × object_height_cm, signed (negative = inverted)."},{"name":"image_type","label":"Image type","type":"string","decimals":4,"description":"real (light actually converges there, can be projected on a screen) or virtual (appears to come from that point)."},{"name":"orientation","label":"Orientation","type":"string","decimals":4,"description":"inverted (m < 0) or upright (m > 0) relative to the object."},{"name":"size","label":"Size","type":"string","decimals":4,"description":"enlarged (|m| > 1), reduced (|m| < 1) or same size (|m| = 1)."},{"name":"image_location_note","label":"Image location","type":"string","decimals":4,"description":"Where the image lies and what the sign of image_distance_cm means for this optic."}],"input_schema":{"type":"object","properties":{"optic":{"description":"Type of optic; it sets the sign of the focal length (converging lens and concave mirror positive, diverging lens and convex mirror negative).","type":"string","enum":["converging_lens","diverging_lens","concave_mirror","convex_mirror"],"examples":["converging_lens"]},"focal_length_cm":{"description":"Magnitude of the focal length in centimetres, always positive; the sign is applied from optic. For a spherical mirror f = R / 2. Unit: cm.","type":"number","exclusiveMinimum":0,"examples":[10],"x-unit":"cm"},"object_distance_cm":{"description":"Distance from the object to the lens or mirror in centimetres (a real object, always positive). Unit: cm.","type":"number","exclusiveMinimum":0,"examples":[15],"x-unit":"cm"},"object_height_cm":{"description":"Height of the object in centimetres (default 1, so image_height_cm equals the magnification). Unit: cm.","type":"number","exclusiveMinimum":0,"default":1,"examples":[1],"x-unit":"cm"}},"additionalProperties":false,"required":["optic","focal_length_cm","object_distance_cm"]},"output_schema":{"type":"object","properties":{"focal_length_signed_cm":{"description":"Focal length with the sign convention applied: positive for converging lens and concave mirror, negative for diverging lens and convex mirror. Unit: cm.","type":"number","x-unit":"cm"},"image_distance_cm":{"description":"d_i = 1 / (1/f − 1/d_o), signed: positive = real image, negative = virtual image (see image_location_note). Unit: cm.","type":"number","x-unit":"cm"},"magnification":{"description":"m = −d_i / d_o; negative means inverted, |m| > 1 enlarged.","type":"number"},"image_height_cm":{"description":"m × object_height_cm, signed (negative = inverted). Unit: cm.","type":"number","x-unit":"cm"},"image_type":{"description":"real (light actually converges there, can be projected on a screen) or virtual (appears to come from that point).","type":"string"},"orientation":{"description":"inverted (m < 0) or upright (m > 0) relative to the object.","type":"string"},"size":{"description":"enlarged (|m| > 1), reduced (|m| < 1) or same size (|m| = 1).","type":"string"},"image_location_note":{"description":"Where the image lies and what the sign of image_distance_cm means for this optic.","type":"string"}}},"formula":"1 / f = 1 / object_distance_cm + 1 / image_distance_cm with f = +focal_length_cm (converging lens, concave mirror) or −focal_length_cm (diverging lens, convex mirror); magnification = −image_distance_cm / object_distance_cm; image_height_cm = magnification × object_height_cm","method":"Gaussian thin-lens and mirror equations in the real-is-positive convention (paraxial rays, negligible lens thickness): for lenses a positive image distance lies on the far side of the lens, for mirrors in front of the mirror; an object exactly at the focal point gives an image at infinity and is rejected.","sources":[{"name":"Wikipedia – Thin lens (thin lens equation)","url":"https://en.wikipedia.org/wiki/Thin_lens","type":"reference","retrieved_at":"2026-09-24"},{"name":"Wikipedia – Curved mirror (mirror equation)","url":"https://en.wikipedia.org/wiki/Curved_mirror","type":"reference","retrieved_at":"2026-09-24"},{"name":"HyperPhysics – Thin lens equation","url":"http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenseq.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":"Converging lens f = 10 cm, object at 15 cm","inputs":{"optic":"converging_lens","focal_length_cm":10,"object_distance_cm":15},"expected":{"focal_length_signed_cm":10,"image_distance_cm":30,"magnification":-2,"image_height_cm":-2,"image_type":"real","orientation":"inverted","size":"enlarged"},"url":"https://tttkmbb.com/api/v1/calculate/lens-equation?optic=converging_lens&focal_length_cm=10&object_distance_cm=15"},{"name":"Convex mirror f = 10 cm, object at 20 cm","inputs":{"optic":"convex_mirror","focal_length_cm":10,"object_distance_cm":20},"expected":{"focal_length_signed_cm":-10,"image_distance_cm":-6.6667,"magnification":0.3333,"image_height_cm":0.3333,"image_type":"virtual","orientation":"upright","size":"reduced"},"url":"https://tttkmbb.com/api/v1/calculate/lens-equation?optic=convex_mirror&focal_length_cm=10&object_distance_cm=20"}],"faq":[{"q":"What happens when the object is inside the focal length of a converging lens?","a":"The image becomes virtual, upright and enlarged, as in a magnifying glass: f = 10 cm and d_o = 5 cm give d_i = −10 cm and m = +2."},{"q":"How do I get the focal length of a mirror?","a":"For a spherical mirror f = R / 2, half the radius of curvature; enter the magnitude and pick concave_mirror or convex_mirror to set the sign."},{"q":"Why is the image distance negative?","a":"A negative d_i is a virtual image: for a lens it lies on the same side as the object (seen by looking through the lens), for a mirror behind the mirror surface. Diverging lenses and convex mirrors always give virtual, upright, reduced images of real objects."}],"tags":["thin lens equation","lens calculator","mirror equation","image distance","magnification","focal length"],"related":[{"calculator_id":"snells-law","reason":"Refraction at a single flat surface, the basis of lens action."},{"calculator_id":"wavelength-frequency","reason":"Wavelength and frequency of the light used."}],"links":{"html":"https://tttkmbb.com/physics/lens-equation","markdown":"https://tttkmbb.com/physics/lens-equation.md","json":"https://tttkmbb.com/physics/lens-equation.json","api":"https://tttkmbb.com/api/v1/calculate/lens-equation","schema":"https://tttkmbb.com/api/v1/calculators/lens-equation","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T03:46:19Z","next_actions":[{"tool":"run_calculator","calculator_id":"lens-equation","reason":"Run Thin Lens Equation Calculator with the inputs above."}],"links":{"markdown":"https://tttkmbb.com/physics/lens-equation.md"}}