CVP verified calculation capability

Solid geometry Calculation Capability

Volume, surface and secondary dimensions for20 Euclidean solid models. Strict decimal inputs, one consistent length unit, binary64 normal-range outputs, bounded figures; ellipsoid surface uses spheroid forms or incomplete elliptic integrals.

Math · Geometry Capability ID math.geometry.solid version 1.19.0 CVP VERIFIED CURRENT Core

Model

Sphere

Sphere: V = (4/3)πr³, A = 4πr²
Cylinder: V = πr²h, A = 2πr(r+h)
Cone: V = (1/3)πr²h, l = √(r²+h²)
Frustum: V = (1/3)πh(R² + Rr + r²)
Prism: V = abh, A = 2(ab+ah+bh)
Pyramid: V = abh/3
Tetrahedron: V = a³√2/12, A = a²√3
Octahedron: V = (√2/3)a³, A = 2√3 a²
Icosahedron: V = (5/12)(3+√5)a³, A = 5√3 a²
Dodecahedron: V = (1/4)(15+7√5)a³, A = 3√(25+10√5) a²
Regular prism: V = A h, A = (n s²)/(4 tan(π/n))
Regular pyramid: V = A h/3, A = (n s²)/(4 tan(π/n))
Regular frustum: V = h/3 (A + a + √(Aa))
Ellipsoid: V = 4/3 πabc
Spherical cap: V = (1/3)πh²(3r−h)
Spherical zone: V = (πh/6)(3a² + 3b² + h²)
Spherical wedge: V = (2/3)r³θ (θ in radians)
Spherical lune: A = 2r²θ (θ in radians)
Spherical sector: V = (2/3)πr²h

Inputs / Outputs

Canonical symbols for this capability. Agent schemas remain the contract.

Symbol	Quantity	Unit policy	Role
`r`	length	consistent length unit	input / output
`radius`	length	consistent length unit	input
`h`	length	consistent length unit	input / output
`height`	length	consistent length unit	input / output
`R`	length	consistent length unit	input / output
`major`	length	consistent length unit	input
`major_r`	—	—	input
`major_radius`	—	—	input
`minor`	length	consistent length unit	input
`R_bottom`	—	—	input
`r1`	—	—	input
`R1`	—	—	input
`base`	—	—	input / output
`R_top`	—	—	input
`r2`	—	—	input
`R2`	—	—	input
`top`	—	—	input
`a`	length	consistent length unit	input / output
`length`	—	—	input
`l`	—	—	input
`b`	length	consistent length unit	input / output
`width`	—	—	input
`w`	—	—	input
`depth`	—	—	input
`s`	length	consistent length unit	input / output
`side`	—	—	input
`edge`	—	—	input
`n`	—	—	input / output
`sides`	—	—	input
`s1`	—	—	input
`s_bottom`	—	—	input
`bottom`	—	—	input
`t`	length	consistent length unit	input / output
`s2`	—	—	input
`s_top`	—	—	input
`rx`	—	—	input
`a_axis`	—	—	input
`semi_a`	—	—	input
`ry`	—	—	input
`b_axis`	—	—	input
`semi_b`	—	—	input
`c`	length	consistent length unit	input / output
`rz`	—	—	input
`c_axis`	—	—	input
`semi_c`	—	—	input
`theta`	scalar plane angle	—	input / output
`angle`	scalar plane angle	—	input
`deg`	scalar plane angle	—	input
`theta_deg`	scalar plane angle	—	input
`volume`	volume	input length unit cubed	output
`surface`	area	square of input unit	output
`lateral`	area	square of input unit	output
`slant`	length	consistent length unit	output
`diagonal`	length	consistent length unit	output
`cylinder`	—	—	output
`cube`	—	—	output
`slant_a`	length	consistent length unit	output
`slant_b`	length	consistent length unit	output
`square_base`	—	—	output
`face`	—	—	output
`cap`	—	—	output
`apothem`	length	consistent length unit	output
`apothem_top`	—	—	output
`prism`	—	—	output
`sphere`	—	—	output
`oblate`	—	—	output
`prolate`	—	—	output
`hemisphere`	—	—	output
`symmetric`	—	—	output
`theta_rad`	scalar plane angle	—	output
`area`	area	square of input unit	output

Trusted Math contract

Inspectable numeric contract for this Capability@Version. Not a rewrite of the engine.

Field	Declaration
`numeric_domain`	real
`exactness`	float64
`algorithm`	Dyadic products/sums with final normal-range rounding, scaled radicals, spheroid forms and Carlson RF/RD incomplete elliptic F/E; output binary64 precision
`valid_domain`	20 right Euclidean models; strict decimal numbers/strings, positive normal binary64 dimensions and required nonzero measures; safe integer3<=n<=9007199254740991; ring torus R>r; cap/sector0<h<=2r; angle0<theta<=360 degrees. Ellipsoid min/max axis ratio>=1e-100. Units L, L2, L3.
`singularity_policy`	NUMERIC_UNDERFLOW for raw nonzero text converting to zero; NUMERIC_RANGE for nonnormal inputs/results or ellipsoid ratio outside envelope. Exact endpoints on normalized binary64. Sphere/spheroid equality branches; no snapping near360 degrees.
`branch_policy`	not applicable — real principal value only
`tolerance`	not applicable — closed form; capability Evidence owns comparison, no global default
`convergence`	not applicable — non-iterative
`conditioning`	Thin shapes and near endpoints are input-sensitive; no input uncertainty bound. Elliptic surface numerically evaluated; not exact arithmetic. Figures are bounded and omitted at extreme ratios, small angles or n>128.
`rounding`	ieee754_round_ties_to_even
`overflow`	ieee754_nonfinite_or_coded_error
`reproducibility`	deterministic algorithm + Capability@Version
`independent_oracle`	Independent mpmath800dps full output references plus separate O2 textbook and Legendre-integrand paths
`golden_vectors`	Evidence golden + boundary + property (and published_reference_case when listed)

Assumptions / Validity

Declared scope for this capability. Do not use it outside this domain.

Assumptions

Euclidean 3-space
The Result Card includes an engine-linked 2-D generating section or silhouette of the published lengths. Not a 3-D mesh. Same pack as REST/MCP.
Right solids; not oblique
Ring torus R > r
Not the circle-only page
Not a vector
Not ISO 5167-5
Not a sphere page for ellipsoid
Not CAS

Example

One numerical check. The calculator page holds worked steps and FAQ.

sphere r=1 → V=4π/3

Open this example in the calculator

Trust

CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all20 published modes; full secondary-output review and raw input/range partitions. Computational verification is defined by CVP. Supplemental domain review (internal) and named expert review are defined by the Review policy and do not by themselves determine CVP conformance.

CVP: CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all20 published modes; full secondary-output review and raw input/range partitions· View Manifest · CVP overview · Specification
Verified scope: all20 published modes; full secondary-output review and raw input/range partitions
Assurance: Core
Declared partition coverage: PASS · 21/21 declared partitions (sphere, cylinder, cone, torus, frustum, prism, pyramid, tetrahedron, octahedron, icosahedron, dodecahedron, regular-prism, regular-pyramid, regular-frustum, ellipsoid, spherical-cap, spherical-zone, spherical-wedge, spherical-lune, spherical-sector, invalid-domain)
Calculation version: 1.19.0
Verification: 115/118 calculation tests · 43/43 legacy golden · 25/25 legacy boundary · 47/50 legacy property
Production surface contract: 5/5
Versions: Calculation 1.19.0 · CVP protocol 1.0.0-proposed · Evidence 2026-10-29.math-uuce-fields-v1
Verification revision: 2026-10-29.math-uuce-fields-v1 · 4/4 property · digest de157af6f244
Reference: O1 model · O2 expected_values · O2 numerical_behavior
Interfaces: PASS · UI (SSR) / REST / MCP
ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 2/2 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
CVP suite: 188/188 golden · 27/27 CVP boundary · 96/96 invalid · 4/4 property · 4/4 metamorphic · 8/8 cross-interface · 2/2 cross-calculator · 5/5 CVP contract
Supplemental domain review: Not performed
Named expert review: Not performed
Version lock: Supported (409 VERSION_MISMATCH)
Artifact consistency: Build verified · 1.19.0 (workspace artifacts agree)
Review-policy tier: Candidate — catalog token candidate; supplemental; not CVP status

This calculator CURRENT · Public schema 1.19.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓

Call this capability

Human page, REST, and MCP share the same engine and calculation_version. Build verified hashes workspace artifacts only. After deploy, attest:production GETs canonical URLs twice — the public/cached view an Agent actually hits, and origin — including every /capabilities/{id} HTML page. Public Capability HTML version must match catalog before production attestation passes. Live Trust rows above fetch /.well-known/calculatorx-production-attestation.json and the output schema const. Build hashes: release integrity. That proves currently published page / engine / schema / evidence / MCP agree — not that a historical version is independently immutable. Reference Calculation Standard — Production surface contract 5/5.

curl -sS -X POST 'https://www.calculatorx.com/api/v1/calc/solid-geometry' \
  -H 'Content-Type: application/json' \
  -d '{"inputs":{"mode":"sphere","r":1},"calculation_version":"1.19.0"}'

{
  "inputs": {
    "mode": "sphere",
    "r": 1
  },
  "calculation_version": "1.19.0"
}

POST /api/v1/calc/solid-geometry. This is a POST calculate endpoint — do not GET it.

Full REST contract: REST API docs.

Tool id: solid-geometry
Engine: solid-geometry
REST: POST /api/v1/calc/solid-geometry
MCP: execute_calculation with capability_id: "math.geometry.solid"
Agent input schema: Canonical (MCP / Agent)
REST input schema: Compatibility (tol[] / tol1…)
Output schema: JSON Schema (latest)
Versioned input schema: Immutable pin (1.19.0)
Versioned output schema: Immutable pin (1.19.0)

References

Human-readable sources plus CVP reference evidence classes.

Sources

Reference evidence classes: O1 model · O2 expected_values · O2 numerical_behavior

Try with an agent

Inspect, then execute. Same engine as REST and the human calculator.

curl -sS -X POST 'https://www.calculatorx.com/mcp' \
  -H 'Content-Type: application/json' \
  -H 'MCP-Protocol-Version: 2026-07-28' \
  -H 'Mcp-Method: tools/call' \
  -H 'Mcp-Name: inspect_capability' \
  -d '{"jsonrpc":"2.0","id":1,"method":"tools/call","params":{"name":"inspect_capability","arguments":{"capability_id":"math.geometry.solid"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

curl -sS -X POST 'https://www.calculatorx.com/mcp' \
  -H 'Content-Type: application/json' \
  -H 'MCP-Protocol-Version: 2026-07-28' \
  -H 'Mcp-Method: tools/call' \
  -H 'Mcp-Name: execute_calculation' \
  -d '{"jsonrpc":"2.0","id":2,"method":"tools/call","params":{"name":"execute_calculation","arguments":{"capability_id":"math.geometry.solid","inputs":{"mode":"sphere","r":1},"calculation_version":"1.19.0"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

Pin calculation_version (1.19.0) when the result must stay reproducible.

Inspect the trust tier and required assurance level before execution. Full protocol notes: MCP docs.