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Math calculator

Sphere Calculator

Sphere V = (4/3)πr³ and A = 4πr². Same engine as Solid Geometry. Not the circle-only page. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
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) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Sphere volume and surface from radius.
Scope
Euclidean 3-space
Verification
Engine tested · Specification checked · v1.18.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.18.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-21.solid-cvp-all-modes
Verification revision
2026-09-21.solid-cvp-all-modes · 4/4 property · digest d81459afdb09
Legacy regression
118/118 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
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.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
26/26 golden · 27/27 CVP boundary · 32/32 invalid · 4/4 property · 8/8 cross-interface · 2/2 cross-calculator · 3/3 CVP contract · Manifest
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
43 legacy golden · 25 legacy boundary · legacy regression suite · 26/26 oracle-backed golden · 32/32 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.18.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

VolumeV = (4/3)πr³
SurfaceA = 4πr²
iSame engine as /solid-geometry. Cylinder and cone stay on that hub or their discovery pages. Not spherical geometry on the plane.

How to use

1

Enter r

Radius must be positive. r = 1 is the unit sphere.

2

Read V and A

Unit sphere V = 4π/3, A = 4π.

Example calculations

Common configurations with formula and result.

ϟ

Unit sphere

r = 1

V = (4/3)π
4π/3
ϟ

r = 3

r = 3

V = 36π
36π

Sphere calculator specification

Version 1.18.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A sphere of radius r>0 has volume (4/3)πr³ and surface 4πr². This page is mode=sphere on math.geometry.solid. The 2-D circle stays /calc/math/circle. Not a second engine. Not CAS.
What it calculates
Sphere volume and surface from radius.
Inputs
  • r
Outputs
  • volume
  • surface
Formula
V = (4/3)πr³, A = 4πr²
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.
  • Not the circle-only page
  • Not CAS
Units
  • length (consistent units)
  • volume (length³)
  • area (length²)
Boundary conditions
  • r ≤ 0 → VALUE_MUST_BE_POSITIVE
  • unknown mode → INVALID_MODE
Example
r=1 → V=4π/3
Validation cases

2 published on this page · 118/118 tests · Production surface contract 5/5 · View evidence

  • r=1 → V=4π/3 A=4π
  • r=3 → V=36π
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.18.0

Other calculators in this family: Cone Calculator, Cylinder Calculator, Dodecahedron Calculator, Ellipsoid Calculator, Frustum Calculator, Icosahedron Calculator, Octahedron Calculator, Prism Calculator . Explore all Solid Geometry.

→ Cone Calculator Right circular cone V = (1/3)πr²h and slant l = √(r²+h²). Same engine as Solid Geometry. Not ISO 5167-5. Not CAS. Runs locally. › → Cylinder Calculator Right circular cylinder V = πr²h and A = 2πr(r+h). Same engine as Solid Geometry. Not CAS. Runs locally. › → Dodecahedron Calculator Regular dodecahedron V = (1/4)(15+7√5)a³ and A = 3√(25+10√5) a² from one edge. Same engine as Solid Geometry. Not an icosahedron. Not an n-gon prism. Not CAS. Runs locally. › → Ellipsoid Calculator Ellipsoid V = 4/3 πabc and surface from the exact elliptic-integral model, numerically evaluated. a = b = c is a sphere. Same engine as Solid Geometry. Not /sphere. Not the 2-D ellipse. Not CAS. Runs locally. › → Frustum Calculator Right circular frustum V = (1/3)πh(R²+Rr+r²) from two radii and height. Same engine as Solid Geometry. Not /cone. Not ISO 5167-5. Not CAS. Runs locally. › → Icosahedron Calculator Regular icosahedron V = (5/12)(3+√5)a³ and A = 5√3 a² from one edge. Same engine as Solid Geometry. Not an octahedron. Not a dodecahedron. Not CAS. Runs locally. › → Octahedron Calculator Regular octahedron V = (√2/3)a³ and A = 2√3 a² from one edge. Same engine as Solid Geometry. Not a tetrahedron. Not an icosahedron. Not CAS. Runs locally. › → Prism Calculator Right rectangular prism V = abh from three edge lengths. Same engine as Solid Geometry. Not a frustum. Not a pyramid. Not CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second solid engine?

No. It is mode=sphere on math.geometry.solid, the same engine as /calc/math/solid-geometry.

Is this the circle calculator?

No. Diameter, circumference, arc, and sector stay on /calc/math/circle.

Where does this run?

Locally in the browser by default. REST and MCP call the same solid-geometry engine.