HomeCalculatorsMathGeometryRegular Pyramid Calculator
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 Engine tested · Specification checked · 66/66 tests · Production surface contract 3/3 · v1.11.0
Input interpretation
Enter values to calculate.
Result
—
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 · 66/66 tests · Production surface contract 3/3 · Specification checked · v1.11.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
25 golden · 13 boundary · 28 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.11.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

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 1.11.0 · 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 · 66/66 tests · Production surface contract 3/3 · 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
1.11.0

Other calculators in this family: Angle Between Lines Calculator, Circle Calculator, Cone Calculator, Conic Section Calculator, Coordinate Geometry Calculator, Cross Product Calculator, Cylinder Calculator, Dodecahedron Calculator . Explore all Geometry.

→ Angle Between Lines Calculator Acute angle between two plane lines, each given by two points. Parallel is 0°. Perpendicular is 90°. Degenerate points are invalid. Same engine as Coordinate Geometry. Not a second engine. Not CAS. Runs locally. › → Circle Calculator Compute diameter, circumference, and area from r, or arc and sector from a central angle. Not the Pythagorean page. Runs locally. › → 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. › → Conic Section Calculator Axis-aligned ellipse, parabola, and hyperbola in standard form. Discovery /ellipse, /parabola, /hyperbola. Not the circle-only page. Not coordinate geometry. Not CAS. Runs locally. › → Coordinate Geometry Calculator Distance, midpoint, and slope of two points in the plane. Line through two points is /line-equation. Point-to-line distance is /point-line-distance. Intersection is /line-intersection. Angle is /angle-between-lines. Not the Pythagorean-only page. Runs locally. › → Cross Product Calculator Vector product u × v in R³. 2-D uses uz = vz = 0. Same engine as Vector. Not a matrix. 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. ›
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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.