HomeCalculatorsMathSolid GeometrySolid Geometry Calculator
Math calculator

Solid Geometry Calculator

Calculate measures for 20 solid models with consistent units. See formulas, numerical limits and a section diagram, then share or export complete results.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all20 published modes; full secondary-output review and raw input/range partitions
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all20 published modes; full secondary-output review and raw input/range partitions
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 · v1.19.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all20 published modes; full secondary-output review and raw input/range partitions· View Manifest · CVP overview · Specification
Versions
Calculation 1.19.0 · CVP protocol 1.0.0-proposed · Evidence 2026-10-29.math-uuce-fields-v1
Verification revision
2026-10-29.math-uuce-fields-v1 · 4/4 property · digest de157af6f244
Legacy regression
115/118 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
188/188 golden · 27/27 CVP boundary · 96/96 invalid · 4/4 property · 4/4 metamorphic · 8/8 cross-interface · 2/2 cross-calculator · 5/5 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 · 188/188 oracle-backed golden · 96/96 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.19.0 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 1.19.0 · 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 the regular-polygon identity. 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 · 115/118 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: Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.
  • 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
  • Euclid’s Elements — XIII.13: Tetrahedron
    Supports: Construction and edge-to-circumsphere geometry of the regular tetrahedron. Face areas are obtained from triangle or regular-pentagon geometry; volume is the sum of pyramids from the center to the faces. These area and volume expressions are derived from the construction and the separately cited pyramid-volume method. Mathematical reference only; not validation or endorsement of this implementation.
  • Euclid’s Elements — XIII.14: Octahedron
    Supports: Construction and edge-to-circumsphere geometry of the regular octahedron. Face areas are obtained from triangle or regular-pentagon geometry; volume is the sum of pyramids from the center to the faces. These area and volume expressions are derived from the construction and the separately cited pyramid-volume method. Mathematical reference only; not validation or endorsement of this implementation.
  • Euclid’s Elements — XIII.16: Icosahedron
    Supports: Construction and edge-to-circumsphere geometry of the regular icosahedron. Face areas are obtained from triangle or regular-pentagon geometry; volume is the sum of pyramids from the center to the faces. These area and volume expressions are derived from the construction and the separately cited pyramid-volume method. Mathematical reference only; not validation or endorsement of this implementation.
  • Euclid’s Elements — XIII.17: Dodecahedron
    Supports: Construction and edge-to-circumsphere geometry of the regular dodecahedron. Face areas are obtained from triangle or regular-pentagon geometry; volume is the sum of pyramids from the center to the faces. These area and volume expressions are derived from the construction and the separately cited pyramid-volume method. Mathematical reference only; not validation or endorsement of this implementation.
  • NIST DLMF — 19.33 Triaxial Ellipsoids — 19.33(i), equations 19.33.1–19.33.3
    Supports: Ellipsoid volume V = 4πabc/3 and exact surface-area expressions in symmetric and Legendre elliptic integrals. An approximate displayed area, where offered, must be distinguished from these exact identities. Mathematical reference only; not validation or endorsement of this implementation.
  • Gilbert Strang, Calculus — Chapter 8: Applications of the Integral (MIT OpenCourseWare) — §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §8.5 Masses and Moments, exercises 24–29 §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area
    Supports: Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Pappus’s volume and surface-area theorems, with a torus volume exercise. For a disk of radius r whose center is distance R>r from the axis, V = 2π²Rr²; for its boundary circle the surface theorem gives S = 4π²Rr². These expressions apply the stated centroid theorems. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
  • Gilbert Strang, Calculus — Chapter 1: Introduction to Calculus (MIT OpenCourseWare) — §1.5 A Review of Trigonometry
    Supports: Right-triangle definitions of sine, cosine and tangent, the sine/cosine identities, and the law-of-cosines and parallelogram exercises 31–32. Chord length follows by bisecting the radii triangle. Regular-polygon apothems and areas follow by splitting the polygon into equal triangles. The law of sines follows by equating a triangle’s altitude computed from either adjacent side; SSA can give two triangles. These applications are derived from the stated relations. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
  • Gilbert Strang, Calculus — Chapter 14: Multiple Integrals (MIT OpenCourseWare) — §14.4 Cylindrical and Spherical Coordinates
    Supports: Spherical coordinates and volume element ρ² sin(φ) dρ dφ dθ. Restricting azimuth to α gives a wedge α/(2π) of a ball and the corresponding surface lune. Radial integration to r with polar angle φ₀ and h=r(1−cos φ₀) gives spherical-sector volume 2πr²h/3. These formulas are derived applications; cap and cone surfaces use the separate surface reference. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
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
1.19.0

Background

Interpretation and common distinctions.

What does this solid geometry calculator compute?

This calculator finds volumes and surface areas for its supported solids, including spheres, circular solids, prisms, pyramids, regular polyhedra, and portions of a sphere. Select the solid and distinguish total surface area from curved or lateral area.

The published formula or method is 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. The following references support the stated mathematics or method:

  • Gilbert Strang, Calculus — Chapter 8: Applications of the Integral (MIT OpenCourseWare): Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Pappus’s volume and surface-area theorems, with a torus volume exercise. For a disk of radius r whose center is distance R>r from the axis, V = 2π²Rr²; for its boundary circle the surface theorem gives S = 4π²Rr². These expressions apply the stated centroid theorems. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
  • Gilbert Strang, Calculus — Chapter 1: Introduction to Calculus (MIT OpenCourseWare): Right-triangle definitions of sine, cosine and tangent, the sine/cosine identities, and the law-of-cosines and parallelogram exercises 31–32. Chord length follows by bisecting the radii triangle. Regular-polygon apothems and areas follow by splitting the polygon into equal triangles. The law of sines follows by equating a triangle’s altitude computed from either adjacent side; SSA can give two triangles. These applications are derived from the stated relations. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
  • Euclid’s Elements — XIII.13: Tetrahedron: Construction and edge-to-circumsphere geometry of the regular tetrahedron. Face areas are obtained from triangle or regular-pentagon geometry; volume is the sum of pyramids from the center to the faces. These area and volume expressions are derived from the construction and the separately cited pyramid-volume method. Mathematical reference only; not validation or endorsement of this implementation.
  • Euclid’s Elements — XIII.14: Octahedron: Construction and edge-to-circumsphere geometry of the regular octahedron. Face areas are obtained from triangle or regular-pentagon geometry; volume is the sum of pyramids from the center to the faces. These area and volume expressions are derived from the construction and the separately cited pyramid-volume method. Mathematical reference only; not validation or endorsement of this implementation.
  • Euclid’s Elements — XIII.16: Icosahedron: Construction and edge-to-circumsphere geometry of the regular icosahedron. Face areas are obtained from triangle or regular-pentagon geometry; volume is the sum of pyramids from the center to the faces. These area and volume expressions are derived from the construction and the separately cited pyramid-volume method. Mathematical reference only; not validation or endorsement of this implementation.
  • Euclid’s Elements — XIII.17: Dodecahedron: Construction and edge-to-circumsphere geometry of the regular dodecahedron. Face areas are obtained from triangle or regular-pentagon geometry; volume is the sum of pyramids from the center to the faces. These area and volume expressions are derived from the construction and the separately cited pyramid-volume method. Mathematical reference only; not validation or endorsement of this implementation.
  • NIST DLMF — 19.33 Triaxial Ellipsoids: Ellipsoid volume V = 4πabc/3 and exact surface-area expressions in symmetric and Legendre elliptic integrals. An approximate displayed area, where offered, must be distinguished from these exact identities. Mathematical reference only; not validation or endorsement of this implementation.
  • NIST DLMF §19.2 — Elliptic Integral Definitions: Definitions of the incomplete elliptic integrals F and E used in ellipsoid surface formulas.
  • Gilbert Strang, Calculus — Chapter 14: Multiple Integrals (MIT OpenCourseWare): Spherical coordinates and volume element ρ² sin(φ) dρ dφ dθ. Restricting azimuth to α gives a wedge α/(2π) of a ball and the corresponding surface lune. Radial integration to r with polar angle φ₀ and h=r(1−cos φ₀) gives spherical-sector volume 2πr²h/3. These formulas are derived applications; cap and cone surfaces use the separate surface reference. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

Lengths, areas, and volumes have different units. Check whether a reported area is curved/lateral or total and use the shape-specific dimension constraints.

The worked example uses r = 1 with V = (4/3)π. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.geometry.solid v1.19.0 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Numerical scope and complete results

Use one consistent length unit: lengths are in L, surface areas in L² and volumes in L³. Inputs are finite decimal numbers or strings. Positive dimensions and nonzero output measures must fit normal IEEE-754 binary64. NUMERIC_UNDERFLOW rejects raw nonzero text that converts to zero; NUMERIC_RANGE asks you to rescale when a required measure is outside that range. Outputs retain binary64 precision, without a small-value zero clamp. Input rounding affects very thin shapes; this is numerical evaluation, not exact decimal arithmetic. Regular n must be a safe integer from 3 to 9007199254740991. Ellipsoid minimum/maximum semi-axis ratio must be at least 1e-100.

The 2-D drawing shows a section, elevation or face, not a 3-D mesh. Very thin shapes, tiny or almost full angles and polygons above 128 sides omit the drawing while retaining complete numerical results. Platonic height denotes vertex height for a tetrahedron, opposite-vertex distance for an octahedron, and circumsphere diameter for an icosahedron or dodecahedron. A spherical lune is a surface; its auxiliary volume is the associated wedge volume.

Ordinary interactive calculation runs locally. Shared URLs send their parameters to the server; REST and MCP send inputs for execution. Sharing does not recreate a historical engine: a version mismatch shows the current-engine notice. CSV and result JSON include all calculated fields and raw provided inputs.

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 Calculate measures for a right cone with consistent units. See formulas, numerical limits and a section diagram, then share or export complete results. › → Cylinder Calculator Calculate measures for a right cylinder with consistent units. See formulas, numerical limits and a section diagram, then share or export complete results. › → Dodecahedron Calculator Calculate measures for a dodecahedron with consistent units. See formulas, numerical limits and a section diagram, then share or export complete results. › → Ellipsoid Calculator Calculate measures for an ellipsoid with consistent units. See formulas, numerical limits and a section diagram, then share or export complete results. › → Frustum Calculator Calculate measures for a conical frustum with consistent units. See formulas, numerical limits and a section diagram, then share or export complete results. › → Icosahedron Calculator Calculate measures for an icosahedron with consistent units. See formulas, numerical limits and a section diagram, then share or export complete results. › → Octahedron Calculator Calculate measures for an octahedron with consistent units. See formulas, numerical limits and a section diagram, then share or export complete results. › → Prism Calculator Calculate measures for a rectangular prism with consistent units. See formulas, numerical limits and a section diagram, then share or export complete results. ›

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.