# Moment of Inertia API > Rigid-body rotational-inertia mechanics as an API, computed locally and deterministically. The shape endpoint returns the mass moment of inertia and the radius of gyration k = √(I/m) for a named standard body about its characteristic axis — a solid sphere (I = 2/5·m·r²), thin spherical shell (2/3·m·r²), solid cylinder or disk (1/2·m·r²), annular/hollow cylinder (1/2·m·(r1²+r2²)), thin ring (m·r²), thin rod about its centre (1/12·m·l²) or about one end (1/3·m·l²), rectangular plate or cuboid (1/12·m·(a²+b²)), solid cone (3/10·m·r²) and point mass (m·r²) — so a 2 kg solid sphere of radius 0.5 m has I = 0.2 kg·m². The parallel-axis endpoint applies the Steiner theorem I = I_cm + m·d² to shift a moment of inertia from the centre-of-mass axis to any parallel axis a distance d away. The shapes endpoint lists the whole catalog with its formulas. All quantities are SI (kg, m → kg·m²). Everything is computed locally and deterministically, so it is instant and private. Ideal for mechanical-engineering, robotics, CAD/CAE, rotating-machinery, structural-dynamics and physics-education app developers, flywheel-and-shaft design tools, and simulation software. Pure local computation — no key, no third-party service, instant. Live, nothing stored. 3 endpoints. This is rotational inertia; for stored rotational energy and flywheel sizing use a flywheel API and for torque and angular acceleration a torque API. ## Authentication All requests require your oanor API key in the `x-oanor-key` header. Get one at https://www.oanor.com/developer/keys. ```bash curl -H "x-oanor-key: oanor_live_…" "https://api.oanor.com/momentofinertia-api/..." ``` ## Pricing - **Free** (Free) — 3,800 calls/Mo, 2 req/s - **Starter** ($7/Mo) — 36,000 calls/Mo, 6 req/s - **Pro** ($21/Mo) — 175,000 calls/Mo, 15 req/s - **Mega** ($65/Mo) — 1,080,000 calls/Mo, 40 req/s ## Endpoints ### Inertia #### `GET /v1/parallel-axis` — Parallel-axis (Steiner) theorem **Parameters:** - `i_cm` (query, required, string) — Moment about centre-of-mass axis (kg·m²) Example: `0.02` - `mass` (query, required, string) — Mass (kg) Example: `1` - `distance` (query, required, string) — Distance d to parallel axis (m) Example: `0.3` **Example:** ```bash curl -H "x-oanor-key: $KEY" \ "https://api.oanor.com/momentofinertia-api/v1/parallel-axis?i_cm=0.02&mass=1&distance=0.3" ``` **Response:** ```json { "data": { "note": "Parallel-axis (Steiner) theorem: I = I_cm + m·d², where I_cm is the moment about the centre-of-mass axis and d is the perpendicular distance to the parallel axis.", "inputs": { "i_cm": 0.02, "mass": 1, "distance": 0.3 }, "shift_term": 0.09, "moment_of_inertia": 0.11 }, "meta": { "timestamp": "2026-06-05T19:50:31.342Z", "request_id": "d2177c8d-c73c-4170-ba8c-1749d3615044" }, "status": "ok", "message": "Parallel-axis theorem", "success": true } ``` #### `GET /v1/shape` — Moment of inertia of a shape **Parameters:** - `shape` (query, required, string) — Shape name (see /v1/shapes) Example: `solid_sphere` - `mass` (query, required, string) — Mass (kg) Example: `2` - `r` (query, optional, string) — Radius (m) Example: `0.5` - `l` (query, optional, string) — Length (m) - `a` (query, optional, string) — Side a (m) - `b` (query, optional, string) — Side b (m) - `r1` (query, optional, string) — Inner radius (m) - `r2` (query, optional, string) — Outer radius (m) **Example:** ```bash curl -H "x-oanor-key: $KEY" \ "https://api.oanor.com/momentofinertia-api/v1/shape?shape=solid_sphere&mass=2&r=0.5" ``` **Response:** ```json { "data": { "note": "Moment of inertia in kg·m². Radius of gyration k = √(I/m). Use /v1/parallel-axis to shift to a parallel axis.", "inputs": { "r": 0.5, "mass": 2, "shape": "solid_sphere" }, "formula": "I = 2/5·m·r²", "moment_of_inertia": 0.2, "radius_of_gyration": 0.316227766 }, "meta": { "timestamp": "2026-06-05T19:50:31.470Z", "request_id": "3a9e243d-8d2e-4064-8716-34a45c5bf11b" }, "status": "ok", "message": "Moment of inertia of a shape", "success": true } ``` #### `GET /v1/shapes` — Shape catalog **Example:** ```bash curl -H "x-oanor-key: $KEY" \ "https://api.oanor.com/momentofinertia-api/v1/shapes" ``` **Response:** ```json { "data": { "note": "All masses in kg, all lengths/radii in m, result in kg·m². Pass shape + mass + the listed params to /v1/shape.", "count": 12, "shapes": { "rod_end": { "params": [ "mass", "l" ], "formula": "I = 1/3·m·l² (thin rod about one end, ⊥)" }, "thin_ring": { "params": [ "mass", "r" ], "formula": "I = m·r² (hoop about central axis)" }, "point_mass": { "params": [ "mass", "r" ], "formula": "I = m·r² (point mass at distance r)" }, "rod_center": { "params": [ "mass", "l" ], "formula": "I = 1/12·m·l² (thin rod about centre, ⊥)" }, "solid_cone": { "params": [ "mass", "r" ], "formula": "I = 3/10·m·r² (about central axis)" }, "solid_disk": { "params": [ "mass", "r" ], "formula": "I = 1/2·m·r² (about central axis)" }, "solid_cuboid": { " …(truncated, see openapi.json for full schema) ``` ### Meta #### `GET /v1/meta` — Spec **Example:** ```bash curl -H "x-oanor-key: $KEY" \ "https://api.oanor.com/momentofinertia-api/v1/meta" ``` **Response:** ```json { "data": { "notes": "SI units (kg, m → kg·m²). This is rigid-body rotational inertia; for stored rotational energy and flywheels use a flywheel API.", "service": "momentofinertia-api", "endpoints": { "GET /v1/meta": "This document.", "GET /v1/shape": "Moment of inertia and radius of gyration for a named shape.", "GET /v1/shapes": "Catalog of supported shapes and their formulas.", "GET /v1/parallel-axis": "Shift a moment of inertia to a parallel axis (I = I_cm + m·d²)." }, "description": "Moment of inertia of standard rigid bodies, the parallel-axis (Steiner) theorem, and the radius of gyration." }, "meta": { "timestamp": "2026-06-05T19:50:31.653Z", "request_id": "bdd61033-9366-4537-a65a-d67c36e37e9c" }, "status": "ok", "message": "Meta", "success": true } ``` --- Marketplace page: https://www.oanor.com/api/momentofinertia-api OpenAPI spec: https://www.oanor.com/api/momentofinertia-api/openapi.json