HomeCalculatorsMathSolid GeometrySolid Geometry Calculator
Math calculator

Solid Geometry Calculator

Right circular sphere, cylinder, cone, ring torus, conical frustum, right rectangular prism, right rectangular pyramid, regular tetrahedron, regular octahedron, regular icosahedron, regular dodecahedron, regular n-gon prism, regular n-gon pyramid, regular n-gon frustum, ellipsoid, spherical cap, spherical zone, spherical wedge, spherical lune, and spherical sector. Discovery /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. 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 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 sphere, cylinder, cone, ring torus, conical frustum, right rectangular prism, right rectangular pyramid, regular tetrahedron, regular octahedron, regular icosahedron, regular dodecahedron, regular n-gon prism, regular n-gon pyramid, regular n-gon frustum, ellipsoid, spherical cap, spherical zone, spherical wedge, spherical lune, and spherical sector volume and surface.
Scope
Euclidean 3-space
Verification
Engine tested · Source 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
Sources
Sources
Implementation conventions
  • Ellipsoid modulus k, not m = k² — Incomplete F(φ,k) and E(φ,k) take elliptic modulus k. Passing m = k² as k is a known library footgun and is regression-tested.
  • Ellipsoid internal axis sort — Human inputs keep labels a, b, c. Surface uses max ≥ mid ≥ min. Permuting axes does not change V or A.
Methods
  • Right solids: closed Euclidean identities in IEEE-754 binary64
  • Ellipsoid exactness: volume closed identity; surface exact elliptic-integral / spheroid model, numerically evaluated
  • Ellipsoid algorithm: sort p≥q≥r; sphere 4πp²; oblate artanh; prolate arcsin; else Carlson RF/RD → F(φ,k), E(φ,k) with modulus k
  • Ellipsoid singularity_policy: a=b=c uses 4πa² so 1/sinφ is never evaluated at φ=0
  • Ellipsoid independent_oracle: 2-D parametric surface quadrature (O2) plus Legendre-integrand F,E for the k vs m convention
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.

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)
PrismV = abh, A = 2(ab+ah+bh)
PyramidV = abh/3
TetrahedronV = a³√2/12, A = a²√3
OctahedronV = (√2/3)a³, A = 2√3 a²
IcosahedronV = (5/12)(3+√5)a³, A = 5√3 a²
DodecahedronV = (1/4)(15+7√5)a³, A = 3√(25+10√5) a²
Regular prismV = A h, A = (n s²)/(4 tan(π/n))
Regular pyramidV = A h/3, A = (n s²)/(4 tan(π/n))
Regular frustumV = h/3 (A + a + √(Aa))
EllipsoidV = 4/3 πabc
Spherical capV = (1/3)πh²(3r−h)
Spherical zoneV = (πh/6)(3a² + 3b² + h²)
Spherical wedgeV = (2/3)r³θ (θ in radians)
Spherical luneA = 2r²θ (θ in radians)
Spherical sectorV = (2/3)πr²h
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, frustum, prism, pyramid, tetrahedron, octahedron, icosahedron, dodecahedron, regular prism, regular pyramid, regular frustum, ellipsoid, spherical cap, spherical zone, spherical wedge, spherical lune, or spherical sector

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. Prism and pyramid need edges a, b, h>0. Tetrahedron, octahedron, icosahedron, and dodecahedron need edge a>0. Regular prism and regular pyramid need integer n≥3 and s, h>0. Regular frustum also needs top side t>0. Ellipsoid needs semi-axes a, b, c>0. Spherical cap and spherical sector need r>0 and 0<h≤2r. Spherical zone needs a, b, h>0. Spherical wedge and spherical lune need r>0 and 0<θ≤360°.

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. Prism 2-3-4 has V=24. Pyramid 2-3-4 has V=8. Unit tetrahedron V=√2/12. Unit octahedron V=√2/3. Unit icosahedron V=(5/12)(3+√5). Unit dodecahedron V=(15+7√5)/4. Unit hexagonal prism V=3√3/2. Unit hexagonal pyramid V=√3/2. Hexagonal frustum 2-1-1 V=7√3/2. Ellipsoid 2-1.5-1 V=4π. Unit hemisphere cap V=2π/3. Symmetric zone 1-1-1 V=7π/6. Unit 90° wedge V=π/3. Unit 90° lune A=π. Unit hemisphere sector V=2π/3.

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
ϟ

Prism 2-3-4

a = 2, b = 3, h = 4

V = 24
24
ϟ

Pyramid 2-3-4

a = 2, b = 3, h = 4

V = 8
8
ϟ

Unit tetrahedron

a = 1

V = √2/12
√2/12
ϟ

Unit octahedron

a = 1

V = √2/3
√2/3
ϟ

Unit icosahedron

a = 1

V = (5/12)(3+√5)
(5/12)(3+√5)
ϟ

Unit dodecahedron

a = 1

V = (15+7√5)/4
(15+7√5)/4
ϟ

Unit hexagonal prism

n = 6, s = 1, h = 1

V = 3√3/2
3√3/2
ϟ

Unit hexagonal pyramid

n = 6, s = 1, h = 1

V = √3/2
√3/2
ϟ

Hexagonal frustum 2-1-1

n = 6, s = 2, t = 1, h = 1

V = 7√3/2
7√3/2
ϟ

Ellipsoid 2-1.5-1

a = 2, b = 1.5, c = 1

V = 4π
4π
ϟ

Symmetric zone 1-1-1

a = 1, b = 1, h = 1

V = 7π/6
7π/6
ϟ

Unit 90° wedge

r = 1, θ = 90°

V = π/3
π/3
ϟ

Unit 90° lune

r = 1, θ = 90°

A = π
π
ϟ

Unit hemisphere sector

r = 1, h = 1

V = 2π/3
2π/3

Solid Geometry calculator specification

Version 1.18.0 · 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). Right rectangular prism V=abh. Right rectangular pyramid V=abh/3 with apex above the base center. Regular tetrahedron V=a³√2/12. Regular octahedron V=(√2/3)a³. Regular icosahedron V=(5/12)(3+√5)a³. Regular dodecahedron V=(1/4)(15+7√5)a³. Regular n-gon prism V=A h with A from math.geometry.plane. Regular n-gon pyramid V=A h/3. Regular n-gon frustum V=h/3(A+a+√(Aa)). Ellipsoid V=4/3 πabc; a=b=c is a sphere. Spherical cap V=(1/3)πh²(3r−h) with 0<h≤2r. Spherical zone V=(πh/6)(3a²+3b²+h²). Spherical wedge V=(2/3)r³θ with θ in radians. Spherical lune A=2r²θ. Spherical sector V=(2/3)πr²h. Not the 2-D circle page. Not ISO 5167-5. Not CAS.
What it calculates
Right circular sphere, cylinder, cone, ring torus, conical frustum, right rectangular prism, right rectangular pyramid, regular tetrahedron, regular octahedron, regular icosahedron, regular dodecahedron, regular n-gon prism, regular n-gon pyramid, regular n-gon frustum, ellipsoid, spherical cap, spherical zone, spherical wedge, spherical lune, and spherical sector volume and surface.
Inputs
  • mode
  • r
  • h?
  • R?
  • a?
  • b?
  • c?
  • n?
  • s?
  • t?
  • theta?
Outputs
  • volume
  • surface
  • area?
  • lateral?
  • slant?
  • slant_a?
  • slant_b?
  • diagonal?
  • height?
  • face?
  • base?
  • cap?
Formula
V = (4/3)πr³; V = πr²h; V = (1/3)πr²h; V = 2π² R r²; V = (1/3)πh(R²+Rr+r²); V = abh; V = abh/3; V = a³√2/12; V = (√2/3)a³; V = (5/12)(3+√5)a³; V = (1/4)(15+7√5)a³; V = A h; V = A h/3; V = h/3 (A+a+√(Aa)); V = 4/3 πabc; V = (1/3)πh²(3r−h); V = (πh/6)(3a²+3b²+h²); V = (2/3)r³θ; A = 2r²θ; V = (2/3)πr²h
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.
  • Right solids; not oblique
  • Ring torus R > r
  • Not the circle-only page
  • Not a vector
  • Not ISO 5167-5
  • Not a sphere page for ellipsoid
  • Not CAS
Units
  • length (consistent units)
  • volume (length³)
  • area (length²)
Boundary conditions
  • r ≤ 0 or h ≤ 0 or R ≤ 0 or a ≤ 0 or b ≤ 0 → VALUE_MUST_BE_POSITIVE
  • torus R ≤ r → INVALID_INPUT
  • frustum R ≤ 0 or r ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • prism a ≤ 0 or b ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • pyramid a ≤ 0 or b ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • tetrahedron a ≤ 0 → VALUE_MUST_BE_POSITIVE
  • octahedron a ≤ 0 → VALUE_MUST_BE_POSITIVE
  • icosahedron a ≤ 0 → VALUE_MUST_BE_POSITIVE
  • dodecahedron a ≤ 0 → VALUE_MUST_BE_POSITIVE
  • regular_prism n < 3 → INVALID_INPUT
  • regular_prism s ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • regular_pyramid n < 3 → INVALID_INPUT
  • regular_pyramid s ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • regular_frustum n < 3 → INVALID_INPUT
  • regular_frustum s ≤ 0 or t ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • ellipsoid a ≤ 0 or b ≤ 0 or c ≤ 0 → VALUE_MUST_BE_POSITIVE
  • spherical_cap h > 2r → INVALID_INPUT
  • spherical_zone a ≤ 0 or b ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • spherical_wedge r ≤ 0 or θ ≤ 0 → VALUE_MUST_BE_POSITIVE
  • spherical_wedge θ > 360° → INVALID_INPUT
  • spherical_lune r ≤ 0 or θ ≤ 0 → VALUE_MUST_BE_POSITIVE
  • spherical_lune θ > 360° → INVALID_INPUT
  • spherical_sector r ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • spherical_sector h > 2r → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
sphere r=1 → V=4π/3
Validation cases

19 published on this page · 118/118 tests · Production surface contract 5/5 · 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
  • mode=prism a=2 b=3 h=4 → V=24, A=52
  • mode=pyramid a=2 b=3 h=4 → V=8
  • mode=tetrahedron a=1 → V=√2/12
  • mode=octahedron a=1 → V=√2/3
  • mode=icosahedron a=1 → V=(5/12)(3+√5)
  • mode=dodecahedron a=1 → V=(15+7√5)/4
  • mode=regular_prism n=6 s=1 h=1 → V=3√3/2
  • mode=regular_pyramid n=6 s=1 h=1 → V=√3/2
  • mode=regular_frustum n=6 s=2 t=1 h=1 → V=7√3/2
  • mode=ellipsoid a=2 b=1.5 c=1 → V=4π A≈27.8864424735
  • mode=spherical_cap r=1 h=1 → V=2π/3 A=3π
  • mode=spherical_zone a=1 b=1 h=1 → V=7π/6
  • mode=spherical_wedge r=1 theta=90 → V=π/3
  • mode=spherical_lune r=1 theta=90 → A=π
  • mode=spherical_sector r=1 h=1 → V=2π/3
Methods
  • Right solids: closed Euclidean identities in IEEE-754 binary64
  • Ellipsoid exactness: volume closed identity; surface exact elliptic-integral / spheroid model, numerically evaluated
  • Ellipsoid algorithm: sort p≥q≥r; sphere 4πp²; oblate artanh; prolate arcsin; else Carlson RF/RD → F(φ,k), E(φ,k) with modulus k
  • Ellipsoid singularity_policy: a=b=c uses 4πa² so 1/sinφ is never evaluated at φ=0
  • Ellipsoid independent_oracle: 2-D parametric surface quadrature (O2) plus Legendre-integrand F,E for the k vs m convention
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Quantity names and symbols for the solid-geometry identities, including ellipsoid volume V = 4/3 πabc
  • NIST DLMF §19.2 Incomplete elliptic integrals — 19.2(ii) Legendre’s integrals F(φ,k) and E(φ,k) · accessed 2026-09-21
    Supports: Ellipsoid triaxial surface uses incomplete F and E with modulus k, not parameter m = k²
  • John D. Cook — Ellipsoid surface area · 2014-07-06 · accessed 2026-09-21
    Supports: Triaxial ellipsoid surface in incomplete F(φ,k), E(φ,k) after sorting a ≥ b ≥ c
Implementation conventions
  • Ellipsoid modulus k, not m = k² — Incomplete F(φ,k) and E(φ,k) take elliptic modulus k. Passing m = k² as k is a known library footgun and is regression-tested.
  • Ellipsoid internal axis sort — Human inputs keep labels a, b, c. Surface uses max ≥ mid ≥ min. Permuting axes does not change V or A.
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 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.