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Math calculator

Regular Prism Calculator

Regular n-gon prism V = A h with A = (n s²)/(4 tan(π/n)) from n, side, and height. Base from Regular Polygon. Same engine as Solid Geometry. Not /prism. Not /regular-pyramid. 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 regular n-gon prism volume, lateral area, and total surface from n, side, and height.
Scope
Right regular n-gon prism
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
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 = A h, A = (n s²)/(4 tan(π/n))
Total surfaceS = 2A + n s h
iSame engine as /solid-geometry. The base area is the same calculation as /regular-polygon. The rectangular prism stays /prism. Regular n-gon pyramid is /regular-pyramid. Not a second engine.

How to use

1

Enter n, s, and h

n must be an integer ≥ 3. Default n = 6, s = 1, h = 1 is the unit hexagonal prism.

2

Read V, A, and S

Unit hexagon prism has V = 3√3/2. Equilateral triangular prism n=3, s=2, h=1 has V = √3. Square n=4, s=2, h=3 matches /prism a=2, b=2, h=3.

Example calculations

Common configurations with formula and result.

ϟ

Unit hexagon

n = 6, s = 1, h = 1

V = 3√3/2
3√3/2
ϟ

Triangular prism

n = 3, s = 2, h = 1

V = √3
√3

Regular Prism calculator specification

Version 1.18.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A right regular n-gon prism with integer n≥3, side s>0, and height h>0 has volume A h, where A = (n s²)/(4 tan(π/n)) is the regular polygon base from math.geometry.plane. Lateral area is n s h. Total surface is 2A + n s h. n=4 matches a square rectangular prism. This page is mode=regular_prism on math.geometry.solid. Not /prism. Not /regular-pyramid. Not CAS.
What it calculates
Right regular n-gon prism volume, lateral area, and total surface from n, side, and height.
Inputs
  • n
  • s
  • h
Outputs
  • volume
  • lateral
  • surface
  • base
Formula
V = A h, A = (n s²)/(4 tan(π/n)), S = 2A + n s h
Assumptions
  • Right regular n-gon prism
  • Integer n ≥ 3
  • Base from math.geometry.plane
  • Not a rectangular prism page
  • Not an n-gon pyramid page
  • Not CAS
Units
  • length (consistent units)
  • volume (length³)
  • area (length²)
Boundary conditions
  • n < 3 or n not integer → INVALID_INPUT
  • s ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • unknown mode → INVALID_MODE
Example
n=6 s=1 h=1 → V=3√3/2
Validation cases

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

  • n=6 s=1 h=1 → V=3√3/2 S=3√3+6
  • n=3 s=2 h=1 → V=√3
  • n=2 s=1 h=1 → INVALID_INPUT
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.
  • 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
    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.
  • 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.
Calculation version
1.18.0

Background

Interpretation and common distinctions.

What does this regular prism calculator compute?

A right regular n-gon prism with integer n≥3, side s>0, and height h>0 has volume A h, where A = (n s²)/(4 tan(π/n)) is the regular polygon base of the regular polygon. Lateral area is n s h. Total surface is 2A + n s h. n=4 matches a square rectangular prism.

The published formula or method is V = A h, A = (n s²)/(4 tan(π/n)), S = 2A + n s h. 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.
  • 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.

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 n = 6, s = 1, h = 1 with V = 3√3/2. 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 v1.18.0 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.

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 a second solid engine?

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

Is this the rectangular prism?

No. The right rectangular box stays on /calc/math/prism. This page is a right prism whose bases are regular n-gons. n=4 with equal s matches a square rectangular prism.

Does this include an n-gon pyramid or an oblique prism?

Regular n-gon pyramid is /calc/math/regular-pyramid on the same engine. The base area reuses /regular-polygon. Cube is already /prism with a=b=h.

Where does this run?

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