Spindle torus calculator

You enter major_radius (the center distance from the torus center to the tube centerline) and minor_radius (the tube cross-section radius) as finite positive lengths with major strictly less than minor so the solid self-intersects; the engine returns both radii, volume, area, a self_intersecting flag, torus_type spindle, and formula strings so a student or a log line can verify the identity without reading the source. When either radius is missing, negative, zero, or non-finite, or when the major radius is not strictly smaller than the minor radius, the tool returns invalid_input instead of a silent NaN. Through the API a successful request costs $0.002; validation failures are not charged.

What a spindle torus calculator returns and why major and minor radii matter

The response always echoes major_radius and minor_radius, reports volume as two pi squared R r squared, area as four pi squared R r, self_intersecting as true, torus_type as spindle, and formula strings for both metrics so a student or a log line can verify the identity without reading the source. Missing major or minor radius, non-finite values, a negative dimension, a non-positive dimension, or a major radius that is not strictly less than the minor radius all raise invalid_input with a message that names the failing condition. That discipline keeps automated tests stable, makes caching safe, and prevents a script from treating a ring torus as a spindle.

How the closed forms are evaluated: volume, area, and the spindle guard

Volume is the exact product of two, pi squared, major radius, and minor radius squared, which is the solid of revolution formula from Pappus equivalent to integrating the circular cross-section along the major circle of circumference two pi R, and that closed form remains valid in the spindle regime even though the solid self-intersects. Surface area multiplies four, pi squared, major radius, and minor radius, equal to the product of the minor circumference two pi r and the major circumference two pi R. Optional precision only rounds the published fields; the internal solve always uses full double precision before that final rounding step. The engine resolves major radius from the field major_radius with aliases R, center_radius, and major, and minor radius from minor_radius with aliases r, tube_radius, and minor, requiring each to be finite. Before any product is formed it rejects a negative major_radius or minor_radius with an explicit message that a spindle torus radius cannot be negative, then rejects non-positive values and the non-spindle ordering major_radius greater than or equal to minor_radius, because that configuration is a horn or ring torus rather than the self-intersecting solid people mean by a spindle torus calculator. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline without re-implementing the two-pi-squared volume rule or the spindle ordering check yourself.

Where spindle torus metrics fit in study, design, and code

Students meet the spindle torus when a textbook contrasts ring, horn, and spindle cases from major and minor radii, when an exam demands volume and surface for the self-intersecting solid, or when solid-geometry homework wants both metrics before comparing against the ordinary doughnut formulas. Teachers can pre-generate answer keys that list volume and area for integer major and minor radii with R less than r, and that refuse a negative radius or a non-spindle ordering instead of printing NaN. Mechanical and graphics design sometimes needs the parametric metrics of a self-intersecting tube for mesh generation, collision envelopes, or teaching demos of solids of revolution; knowing that a requested pair is not a spindle is as valuable as the number itself. Surveying and CAD workflows place toroidal volumes as construction geometry, and game or simulation teams use volume and area for procedural mesh sizing that should stay deterministic across platforms when the tube radius exceeds the centerline radius. On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split—free interactive exploration and a priced, identical API—matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result. Pair it with other free geometry endpoints when you need ring-torus volume, cone volume, or sphere surface for a lab of deterministic tools, and with documentation that shows both radii and the finished metrics so the numbers and the picture stay aligned for learners and reviewers. Whether you arrived searching for a spindle torus calculator or for self-intersecting volume and area from major and minor radii, the contract is the same: positive major and minor radii with major strictly less than minor in, finite metrics or a clear invalid_input out.