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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 Engine tested · Specification checked · 16/16 tests · Production surface contract 3/3 · v1.1.0
Input interpretation
Enter values to calculate.
Result
—
Model
Sphere volume and surface from radius.
Scope
Euclidean 3-space
Verification
Engine tested · 16/16 tests · Production surface contract 3/3 · Specification checked · v1.1.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
5 golden · 3 boundary · 8 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.1.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

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 1.1.0 · 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
  • 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 · 16/16 tests · Production surface contract 3/3 · 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
1.1.0

Other calculators in this family: Angle Between Lines Calculator, Circle Calculator, Cone Calculator, Conic Section Calculator, Coordinate Geometry Calculator, Cross Product Calculator, Cylinder Calculator, Dot Product Calculator . Explore all Geometry.

→ Angle Between Lines Calculator Acute angle between two plane lines, each given by two points. Parallel is 0°. Perpendicular is 90°. Degenerate points are invalid. Same engine as Coordinate Geometry. Not a second engine. Not CAS. Runs locally. › → Circle Calculator Compute diameter, circumference, and area from r, or arc and sector from a central angle. Not the Pythagorean page. Runs locally. › → 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. › → Conic Section Calculator Axis-aligned ellipse, parabola, and hyperbola in standard form. Discovery /ellipse, /parabola, /hyperbola. Not the circle-only page. Not coordinate geometry. Not CAS. Runs locally. › → Coordinate Geometry Calculator Distance, midpoint, and slope of two points in the plane. Line through two points is /line-equation. Point-to-line distance is /point-line-distance. Intersection is /line-intersection. Angle is /angle-between-lines. Not the Pythagorean-only page. Runs locally. › → Cross Product Calculator Vector product u × v in R³. 2-D uses uz = vz = 0. Same engine as Vector. Not a matrix. 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. › → Dot Product Calculator Scalar product u · v = ux vx + uy vy + uz vz. Same engine as Vector. Not a matrix. Not CAS. Runs locally. ›
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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.