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See Review policy for supplemental domain assurance." }, "internal_review": { "status": "not_applicable", "reviewer": null }, "page": "https://www.calculatorx.com/calc/math/plane-geometry", "api": "https://www.calculatorx.com/api/v1/calc/plane-geometry", "schema_input": "https://www.calculatorx.com/schemas/math.geometry.plane.input.json", "schema_output": "https://www.calculatorx.com/schemas/math.geometry.plane.output.json", "assumptions": [ "Euclidean plane", "Regular polygon integer n ≥ 3", "Annulus R > r", "Parallelogram θ in (0°, 180°)", "Isosceles trapezoid a, b, h > 0", "Irregular polygon shoelace 3–8 vertices", "Not the circle-only page", "Not a triangle solver", "Not solid geometry", "Not /trapezoidal-rule", "Not the fluid trapezoid", "Not CAS" ], "limitations": [ "Euclidean plane figures only", "Not the circle-only page — 2-D circle stays on math.geometry.circle", "Not a triangle solver — SSS/SAS stay on math.geometry.triangle", "Not solid geometry (sphere / cylinder / cone)", "Regular polygon needs integer n ≥ 3", "Annulus needs outer R > inner r; R = r is a circle, not an annulus", "Parallelogram included angle θ must be in (0°, 180°); θ=90° recovers a rectangle", "Isosceles trapezoid needs bases a,b>0 and height h>0; a=b recovers a rectangle; not the fluid trapezoid; not /trapezoidal-rule", "Irregular polygon uses shoelace on 3–8 vertices; simple polygon assumed; area 0 is invalid; not /regular-polygon; not a triangle solver", "unknown mode (including cas, sphere, pythagorean) → INVALID_MODE" ], "boundary_conditions": [ "a ≤ 0 or b ≤ 0 or s ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE", "n < 3 or non-integer n → INVALID_INPUT", "annulus R ≤ r → INVALID_INPUT", "parallelogram θ ≤ 0° or θ ≥ 180° → INVALID_INPUT", "irregular n < 3 or n > 8 or A=0 → INVALID_INPUT", "unknown mode → INVALID_MODE" ], "sources": [ { "title": "ISO 80000-2:2019 Quantities and units — Mathematics", "url": "https://www.iso.org/standard/64973.html", "supports": "Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.", "kind": "standard", "publisher": "International Organization for Standardization", "reference_scope": "all" }, { "title": "CRC Standard Mathematical Tables — plane polygons", "supports": "Regular n-gon A = (n s²)/(4 tan(π/n)); parallelogram A = ab sin θ; shoelace / Meister–Gauss polygon area", "kind": "reference", "reference_scope": "model", "oracle_class": "O1" }, { "title": "Euclid’s Elements — I.35: Parallelograms on the Same Base", "url": "https://mathcs.clarku.edu/~djoyce/elements/bookI/propI35.html", "supports": "Equal-base, equal-height parallelograms have equal area. Comparing with a rectangle gives A = base × perpendicular height; side length is not generally the height. Mathematical reference only; not validation or endorsement of this implementation.", "kind": "reference", "publisher": "Clark University, David E. Joyce edition" }, { "title": "Gilbert Strang, Calculus — Chapter 9: Polar Coordinates and Complex Numbers (MIT OpenCourseWare)", "section": "§§9.1–9.3 Polar Coordinates, Length and Area", "url": "https://ocw.mit.edu/courses/res-18-001-calculus-fall-2023/mitres_18_001_f17_ch09.pdf", "supports": "Polar coordinates and the circular wedge area r²Δθ/2, together with area = ∫r² dθ/2 and polar arc-length integration. A full turn gives disk area πr² and circumference 2πr. An annulus follows by subtracting disk areas and adding both boundary lengths; these are derived applications. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.", "kind": "reference", "publisher": "Wellesley-Cambridge Press; hosted by MIT OpenCourseWare" }, { "title": "Gilbert Strang, Calculus — Chapter 15: Vector Calculus (MIT OpenCourseWare)", "section": "§15.3 Green’s Theorem", "url": "https://ocw.mit.edu/courses/res-18-001-calculus-fall-2023/mitres_18_001_f17_ch15.pdf", "supports": "The planar area formula A = (1/2) ∮(x dy − y dx). For straight polygon edges it gives the shoelace expression; rectangular/square areas and trapezoid area (B+b)h/2 follow by applying it to the vertices. For x=a cos t, y=b sin t it gives ellipse area πab. Perimeters of these polygons are sums of side lengths. These formulas are derived applications of the theorem. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.", "kind": "reference", "publisher": "Wellesley-Cambridge Press; hosted by MIT OpenCourseWare" }, { "title": "Gilbert Strang, Calculus — Chapter 1: Introduction to Calculus (MIT OpenCourseWare)", "section": "§1.5 A Review of Trigonometry", "url": "https://ocw.mit.edu/courses/res-18-001-calculus-fall-2023/mitres_18_001_f17_ch01.pdf", "supports": "Right-triangle definitions of sine, cosine and tangent, the sine/cosine identities, and the law-of-cosines and parallelogram exercises 31–32. Chord length follows by bisecting the radii triangle. Regular-polygon apothems and areas follow by splitting the polygon into equal triangles. The law of sines follows by equating a triangle’s altitude computed from either adjacent side; SSA can give two triangles. These applications are derived from the stated relations. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.", "kind": "reference", "publisher": "Wellesley-Cambridge Press; hosted by MIT OpenCourseWare" } ], "methods": [], "fingerprint": "sha256:8b1a25bde28dba84322ff335d917d048ff26b6d93b75e2ee6c500c523b181eff", "tier": "engine_tested", "verification": { "tests": { "passed": 25, "total": 25, "golden": 9, "boundary": 5, "property": 11 }, "contract_checks": { "passed": 5, "total": 5 } }, "evidence_core_generated_at": "2026-10-14T05:05:51.930Z", "rendered_at": "2026-10-20T02:41:58.125Z", "release_integrity": { "status": "verified", "scope": "workspace", "production_attestation_required": true, "calculation_version": "1.5.0", "artifacts": { "engine": "1.5.0", "page": "1.5.0", "capability": "1.5.0", "evidence": "1.5.0", "input_schema": "1.5.0", "output_schema": "1.5.0", "mcp_contract": "1.5.0" }, "digests": { "input_schema_sha256": "5039fac5be916efbd613bac465d29e64c0e098f17efb094b415669d76b6ace36", "output_schema_sha256": "4f46ff37a1617bce5f50363a965f32b87ec3740b942f35bfdb40fa31ca592515", "evidence_sha256": "e445580c2a7b7baeb2f65c0b1966c6232ac73450cd296665a114f52db87d6e5b", "evidence_core_sha256": "e445580c2a7b7baeb2f65c0b1966c6232ac73450cd296665a114f52db87d6e5b", "capability_sha256": "62fdec73bbc2b8084df33d4ae00e1faed8f01200cc2961f40e8848e1b7da970a", "engine_sha256": "6ef38e64f6a7018cce0538e78ba57c8e2a141f11e8779e9e3f91408a668b7b71", "mcp_contract_sha256": "0f93a3901b1daef0cbb79d9604082f1726731a3b0355aa6a554e550d4fa77116" }, "contract_checks": { "total": 5, "passed": 5, "failed": 0, "items": [ { "id": "release-artifacts-aligned", "status": "pass", "detail": "1.5.0" }, { "id": "schema-output-required-declared", "status": "pass" }, { "id": "schema-valid-input-executes", "status": "pass" }, { "id": "engine-success-output-validates-schema", "status": "pass" }, { "id": "rest-success-payload-validates-schema", "status": "pass" } ] }, "report": "https://www.calculatorx.com/.well-known/calculatorx-release-integrity.json" }, "tests": [], "disclosure": { "policy": "calculatorx-public-evidence-v1", "full_validation": "private", "representation": "summary" } }