HomeCalculatorsMathGeometryPyramid Calculator
Math calculator

Pyramid Calculator

Right rectangular pyramid V = abh/3 from three lengths. Same engine as Solid Geometry. Not a prism. 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 rectangular pyramid volume and surface from base edges and height.
Scope
Right rectangular 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Embedded snapshot: unpublished · Build schema 1.11.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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 = abh/3
Total surfaceA = ab + a√(h²+(b/2)²) + b√(h²+(a/2)²)
iSame engine as /solid-geometry. The matching rectangular box is /prism. The regular tetrahedron is /tetrahedron. Not a second engine.

How to use

1

Enter a, b, and h

Base edges and height must be positive. Default 2, 3, 4. Equal base edges a=b are a valid square pyramid.

2

Read V and A

2-3-4 has volume 8, one third of the 2-3-4 prism. Square 6-6-4 has face slants 5, V=48, A=96.

Example calculations

Common configurations with formula and result.

ϟ

2-3-4

a = 2, b = 3, h = 4

V = 2×3×4/3
8
ϟ

Square 6-6-4

a = 6, b = 6, h = 4

V = 48, A = 96, l = 5
48

Pyramid calculator specification

Version 1.11.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A right rectangular pyramid of base edges a>0, b>0 and height h>0, with the apex above the base center, has volume abh/3. Face altitudes are slant_a=√(h²+(b/2)²) and slant_b=√(h²+(a/2)²). Lateral area is a·slant_a + b·slant_b. Total surface is lateral + ab. a=b is a square base. This page is mode=pyramid on math.geometry.solid. Not a prism. Not a tetrahedron. Not CAS.
What it calculates
Right rectangular pyramid volume and surface from base edges and height.
Inputs
  • a
  • b
  • h
Outputs
  • volume
  • lateral
  • surface
  • slant_a
  • slant_b
Formula
V = abh/3, A = ab + a√(h²+(b/2)²) + b√(h²+(a/2)²)
Assumptions
  • Right rectangular pyramid
  • Apex above the base center
  • a=b is a valid square base
  • Not a prism
  • Not a tetrahedron
  • Not CAS
Units
  • length (consistent units)
  • volume (length³)
  • area (length²)
Boundary conditions
  • a ≤ 0 or b ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • unknown mode → INVALID_MODE
Example
a=2 b=3 h=4 → V=8
Validation cases

3 published on this page · 66/66 tests · Production surface contract 3/3 · View evidence

  • a=2 b=3 h=4 → V=8
  • a=6 b=6 h=4 → V=48 A=96 slant=5 square_base=true
  • a=2 b=3 h=0 → VALUE_MUST_BE_POSITIVE
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=pyramid on math.geometry.solid, the same engine as /calc/math/solid-geometry.

Is this a prism?

No. The right rectangular box stays on /calc/math/prism. This pyramid volume is one third of that prism for the same a, b, and h.

Does this include a tetrahedron or a regular n-gon pyramid?

The regular tetrahedron is /calc/math/tetrahedron. Regular n-gon pyramids are /calc/math/regular-pyramid on the same engine.

Where does this run?

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