#torus

Advanced 3D-solid geometry as an API, computed locally and deterministically — the shapes a basic geometry calculator leaves out. The cone-frustum endpoint gives the volume V = (π·h/3)·(R² + R·r + r²), the slant height √(h² + (R−r)²) and the lateral and total surface area of a truncated cone, the shape of buckets, lampshades and hoppers. The torus endpoint gives a doughnut’s volume 2π²·R·r² and surface area 4π²·R·r from its centre-to-tube and tube radii. The ellipsoid endpoint gives the exact volume (4/3)π·a·b·c and a Knud-Thomsen surface-area approximation accurate to better than 1.1 %. The platonic endpoint returns the volume and surface area of any of the five Platonic solids — tetrahedron, cube, octahedron, dodecahedron and icosahedron — from the edge length, using the exact golden-ratio coefficients (a unit icosahedron has volume 2.1817 and surface area 8.6603). Use a consistent length unit and get area and volume out. Everything is computed locally and deterministically, so it is instant and private. Ideal for engineering, CAD, 3D-modelling, architecture, manufacturing and maths-education app developers, volume-and-area and packaging tools, and simulation software. Pure local computation — no key, no third-party service, instant. Live, nothing stored. 4 endpoints. These are the advanced solids; for sphere, cube, cylinder, cone and 2D shapes use a general geometry API.

api.oanor.com/solids-api