Prolate spheroid calculator

This capability packages both results as deterministic JSON with no computer-algebra system, no random seeds, and the same pure solve function in the free browser tool and the prepaid API path. If the equatorial radius exceeds the polar radius the input is rejected with invalid_input, because that geometry is oblate rather than prolate and belongs to a different search intent and formula set.

How to call the API and what comes back

Use it from geometry homework pipelines, packaging and sports-ball takeoff scripts, planetary figure checks, interview-prep sandboxes, CAD pre-checks, or any workflow that needs reproducible prolate numbers without installing a computer-algebra package on every machine. Send a JSON body with equatorial_radius and polar_radius. Field equatorial_radius is the positive equatorial semi-axis a; aliases a, equatorial, a_eq, and radius_eq are accepted. Field polar_radius is the polar semi-axis c; aliases c, polar, a_polar, and radius_polar map to the same meaning. Both must be finite numbers strictly greater than zero and inside the published limits, share the same length unit, and satisfy polar_radius greater than or equal to equatorial_radius. On success the response echoes both radii; returns volume as the exact four thirds pi a squared c value; returns surface_area from the closed eccentricity formula or the sphere limit; returns eccentricity e (zero on a sphere); flags is_sphere when the two radii are equal; sets method to prolate_spheroid; and includes a short formula block documenting both expressions. Failed validation—missing a required field, a non-finite number, a non-positive radius, an out-of-range magnitude, or equatorial exceeding polar—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item. Pair the call with unit conversion only when your source data mixes millimetres and metres; this capability does not convert linear units for you.

Practical tips, limits, and related searches

When teaching, prefer paired examples: equatorial two and polar four for a clearly elongated solid; equatorial one and polar one for a unit sphere where volume is four thirds pi and surface is four pi; and equatorial three with polar two to demonstrate the hard rejection path when the equatorial measure exceeds the polar one. Prefer this tool when you already know the two prolate radii and need exact volume plus exact surface in one shot. Round-trip checks are easy: after solving a sphere case, confirm volume equals four thirds pi a cubed and surface equals four pi a squared; for unequal radii, confirm volume scales with a squared times c when you double the polar axis. If you need three unequal semi-axes, use the triaxial ellipsoid volume calculator; if you need a pure sphere from one radius, use the dedicated sphere tools instead of forcing equal prolate radii. Keep each radius inside the published positive range up to one trillion so intermediate products and inverse sines stay finite in IEEE double precision on the Worker and in the browser. Because volume only needs a squared times c, students can recompute by hand for nice fractions of pi and match the API line by line after dividing reported floats by pi if desired. Surface checks against a sphere are the best sanity test: set both radii equal and confirm surface_area matches four pi a squared within floating tolerance. This landing targets people searching for a prolate spheroid calculator or volume and surface area from equatorial and polar radii; it is not a general triaxial ellipsoid tool, not an oblate calculator, and not a Knud Thomsen approximation path. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when users type impossible measures while learning the contract.