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Math calculator

Regular Pyramid Calculator

Regular n-gon pyramid V = A h/3 with A = (n s²)/(4 tan(π/n)) from n, side, and height. Base from Regular Polygon. Same engine as Solid Geometry. Not /pyramid. Not a tetrahedron. 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 pyramid volume, lateral area, and total surface from n, side, and height.
Scope
Right regular n-gon pyramid
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 = A h/3, A = (n s²)/(4 tan(π/n))
Total surfaceS = A + n s l/2, l = √(h²+r²)
iSame engine as /solid-geometry. The base area is the same calculation as /regular-polygon. Volume is one third of /regular-prism for the same n, s, and h. The rectangular pyramid stays /pyramid. The regular tetrahedron stays /tetrahedron. 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 pyramid.

2

Read V, A, and S

Unit hexagon pyramid has V = √3/2, one third of the unit hexagonal prism. Square n=4, s=6, h=4 matches /pyramid a=6, b=6, h=4 with V=48 and A=96.

Example calculations

Common configurations with formula and result.

ϟ

Unit hexagon

n = 6, s = 1, h = 1

V = √3/2
√3/2
ϟ

Square pyramid

n = 4, s = 6, h = 4

V = 48, A = 96
48

Regular Pyramid calculator specification

Version 1.18.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A right regular n-gon pyramid with integer n≥3, side s>0, and height h>0, apex above the base center, has volume A h/3, where A = (n s²)/(4 tan(π/n)) is the regular polygon base from math.geometry.plane. Face slant is l=√(h²+r²) with r the base apothem. Lateral area is n s l/2. Total surface is A + n s l/2. n=4 matches a square rectangular pyramid. This page is mode=regular_pyramid on math.geometry.solid. Not /pyramid. Not a tetrahedron. Not CAS.
What it calculates
Right regular n-gon pyramid volume, lateral area, and total surface from n, side, and height.
Inputs
  • n
  • s
  • h
Outputs
  • volume
  • lateral
  • surface
  • base
  • slant
Formula
V = A h/3, A = (n s²)/(4 tan(π/n)), l = √(h²+r²), S = A + n s l/2
Assumptions
  • Right regular n-gon pyramid
  • Apex above the base center
  • Integer n ≥ 3
  • Base from math.geometry.plane
  • Not a rectangular pyramid page
  • Not a tetrahedron
  • 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=√3/2
Validation cases

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

  • n=6 s=1 h=1 → V=√3/2
  • n=4 s=6 h=4 → V=48 A=96
  • n=2 s=1 h=1 → INVALID_INPUT
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, 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_pyramid on math.geometry.solid, the same engine as /calc/math/solid-geometry.

Is this the rectangular pyramid?

No. The right rectangular pyramid stays on /calc/math/pyramid. This page is a right pyramid whose base is a regular n-gon. n=4 with equal s matches a square rectangular pyramid.

Is n=3 a tetrahedron?

No. A triangular n-gon pyramid has an arbitrary height. The regular tetrahedron stays on /calc/math/tetrahedron. Regular n-gon frustum is /calc/math/regular-frustum. The base area reuses /regular-polygon.

Where does this run?

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