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Math calculator

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.

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
Right circular frustum volume, slant, and surface from two radii and height.
Scope
Right circular frustum
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 = (1/3)πh(R² + Rr + r²)
Slantl = √(h² + (R − r)²)
iSame engine as /solid-geometry. The full cone is /cone. The ISO 5167-5 cone meter is /calc/physics/iso-cone. Not a second engine.

How to use

1

Enter R, r, and h

Both radii and the height must be positive. Default R=2, r=1, h=3. Equal radii r=R are a valid cylinder.

2

Read V, l, and A

2-1-3 has volume 7π. Equal radii 1 and 1 with h=1 match the unit cylinder.

Example calculations

Common configurations with formula and result.

ϟ

2-1-3

R = 2, r = 1, h = 3

V = π×3×(4+2+1)/3
7π
ϟ

Cylinder identity

R = 1, r = 1, h = 1

V = π
π

Frustum calculator specification

Version 1.18.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A right circular frustum of bases R>0, r>0 and height h>0 has volume (1/3)πh(R²+Rr+r²) and slant √(h²+(R−r)²). r=R recovers a cylinder. This page is mode=frustum on math.geometry.solid. Not /cone. Not ISO 5167-5. Not CAS.
What it calculates
Right circular frustum volume, slant, and surface from two radii and height.
Inputs
  • R
  • r
  • h
Outputs
  • volume
  • slant
  • lateral
  • surface
Formula
V = (1/3)πh(R²+Rr+r²), l = √(h²+(R−r)²)
Assumptions
  • Right circular frustum
  • r=R is a valid cylinder
  • Not /cone
  • Not ISO 5167-5
  • Not /prism
  • Not a pyramid
  • Not CAS
Units
  • length (consistent units)
  • volume (length³)
  • area (length²)
Boundary conditions
  • R ≤ 0 or r ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • unknown mode → INVALID_MODE
Example
R=2 r=1 h=3 → V=7π
Validation cases

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

  • R=2 r=1 h=3 → V=7π
  • R=1 r=1 h=1 → V=π cylinder=true
  • R=2 r=1 h=0 → VALUE_MUST_BE_POSITIVE
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, Icosahedron Calculator, Octahedron Calculator, Prism Calculator, Pyramid 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. › → 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. › → Pyramid Calculator Right rectangular pyramid V = abh/3 from three lengths. Same engine as Solid Geometry. Not a prism. Not a tetrahedron. Not CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second solid engine?

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

Is this the cone page?

No. The right circular cone stays on /calc/math/cone. Equal radii here recover a cylinder, not a cone.

Is this the ISO 5167-5 cone meter?

No. That meter is /calc/physics/iso-cone. This page is a Euclidean solid of revolution. Regular n-gon frustum is /calc/math/regular-frustum.

Where does this run?

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