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Math calculator

Solid Geometry Calculator

Right circular sphere, cylinder, cone, ring torus, and conical frustum. Discovery /sphere, /cylinder, /cone, /frustum. Not the circle-only page. Not a vector. Not ISO 5167-5. 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
Right circular sphere, cylinder, cone, ring torus, and conical frustum volume and surface.
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Embedded snapshot: unpublished · Build schema 1.1.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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.

SphereV = (4/3)πr³, A = 4πr²
CylinderV = πr²h, A = 2πr(r+h)
ConeV = (1/3)πr²h, l = √(r²+h²)
FrustumV = (1/3)πh(R² + Rr + r²)
iTorus is a ring torus (R>r). Horn and spindle tori are out of scope. Discovery pages lock one mode. Not CAS.

How to use

1

Choose sphere, cylinder, cone, torus, or frustum

Sphere needs r>0. Cylinder and cone also need h>0. Torus needs major R greater than tube r. Frustum needs bases R, r and height h>0.

2

Read volume and surface

Unit sphere V=4π/3. Cone r=3, h=4 has slant 5. Cylinder r=1, h=1 has V=π.

Example calculations

Common configurations with formula and result.

ϟ

Unit sphere

r = 1

V = (4/3)π
4π/3
ϟ

3-4 cone

r = 3, h = 4

l = 5, V = 12π
12π
ϟ

Unit cylinder

r = 1, h = 1

V = π
π
ϟ

Frustum 2-1-3

R = 2, r = 1, h = 3

V = 7π
7π

Solid Geometry calculator specification

Version 1.1.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Right circular solids: sphere V=(4/3)πr³, cylinder V=πr²h, cone V=(1/3)πr²h with slant l=√(r²+h²), ring torus V=2π²Rr² with major R>r, frustum V=(1/3)πh(R²+Rr+r²). Not the 2-D circle page. Not ISO 5167-5. Prism and pyramid come later. Not CAS.
What it calculates
Right circular sphere, cylinder, cone, ring torus, and conical frustum volume and surface.
Inputs
  • mode
  • r
  • h?
  • R?
Outputs
  • volume
  • surface
  • lateral?
  • slant?
Formula
V = (4/3)πr³; V = πr²h; V = (1/3)πr²h; V = 2π² R r²; V = (1/3)πh(R²+Rr+r²)
Assumptions
  • Euclidean 3-space
  • Right circular solids
  • Ring torus R > r
  • Not the circle-only page
  • Not a vector
  • Not ISO 5167-5
  • Not prism / pyramid
  • Not CAS
Units
  • length (consistent units)
  • volume (length³)
  • area (length²)
Boundary conditions
  • r ≤ 0 or h ≤ 0 or R ≤ 0 → VALUE_MUST_BE_POSITIVE
  • torus R ≤ r → INVALID_INPUT
  • frustum R ≤ 0 or r ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • unknown mode → INVALID_MODE
Example
sphere r=1 → V=4π/3
Validation cases

4 published on this page · 16/16 tests · Production surface contract 3/3 · View evidence

  • mode=sphere r=1 → V=4π/3, A=4π
  • mode=cone r=3 h=4 → l=5, V=12π
  • mode=cylinder r=1 h=1 → V=π
  • mode=frustum R=2 r=1 h=3 → V=7π
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 the circle calculator?

No. The 2-D circle page is /calc/math/circle. This engine is 3-D volume and surface.

Is /cone the ISO 5167-5 cone meter?

No. The ISO 5167-5 cone meter is /calc/physics/iso-cone. This cone is a right circular solid.

Where does this run?

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