Mirror focal length from radius

Physics students checking ray-diagram worksheets, teachers building answer keys, optical technicians verifying simple spherical blanks, and tutoring platforms that need structured JSON all hit the same relation. This capacity accepts a finite non-zero radius of curvature, returns focal length, mirror type, formula, method, and the sign convention string, and rejects R equals zero with invalid_input so callers never treat a plane-like degenerate geometry as a valid spherical focus.

What f = R/2 means for spherical mirrors and search intent

Delivering radius_of_curvature, focal_length, mirror_type, formula, method, and sign_convention as JSON makes the result easy to paste into notes, diff in automated tests, or reuse across languages without reimplementing the divide. This endpoint does not solve object and image distances; pair it with a mirror-equation tool when you already know f and need di and magnification.

Input contract, sign convention, and how validation works

Zero radius is always rejected: R equals zero is not a valid spherical mirror surface with a defined center of curvature, and f equals R over two would collapse to zero in a way that misleads homework checks. Under the common school sign convention used with the curved-mirror equation elsewhere in this catalogue, positive R means a concave (converging) mirror and yields positive f; negative R means a convex (diverging) mirror and yields negative f. There is no multi-mirror train, no aspheric figure, and no lensmaker path—this is ideal spherical-mirror algebra from a single known radius. POST a JSON body with required radius_of_curvature. Aliases R, radius, and roc map to the same field so forms and short lab notes can pass familiar symbols. Every provided value must be a finite number or a numeric string that parses to a finite number; missing fields, empty strings, non-numeric text, NaN, and Infinity all fail immediately with invalid_input before any result is returned, and failed validation is not charged. After validation, focal_length is computed as radius_of_curvature divided by two and rounded to a stable twelve-decimal precision so the same input always yields the same JSON across JavaScript engines. The response field mirror_type is therefore concave when R is positive and convex when R is negative.

Practical use, limits, pricing, and delivery channels

Use the browser tool when you want an immediate f equals R over two check while grading homework, verifying a lab spherical-mirror blank, converting a manufacturer radius into focal length for a concave shaving-mirror demo, or sanity-checking that a convex security-mirror sample carries a negative focal length under the school convention. Choose signs consistently so positive f feeds concave cases in the mirror equation and negative f feeds convex cases as textbooks expect. There is no language model, no external optics service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials. Each successful request is one billable item at $0.002; failed validation does not consume balance. Results are fully deterministic: the same radius always yields the same focal_length and mirror_type. Wire the endpoint into optics drills, contest answer checkers, radius-to-focus worksheets, or preprocessing steps where you already prefer explicit domain errors—such as rejecting zero radius—over opaque black-box widgets that cannot explain why a bad input was refused.