Pyramid Volume Calculator – All Base Shapes
A pyramid is a three-dimensional solid with a polygonal base and triangular lateral faces that meet at a single point called the apex. Whether you are working with the famous Egyptian square pyramids, a triangular tetrahedron, or an architectural structure with an irregular base, calculating its volume follows the same universal formula. This calculator supports five base types and seven length units, giving you instant results in cubic metres, cubic centimetres, litres, and cubic feet.
The Universal Pyramid Volume Formula
Regardless of the base shape, the volume of any pyramid is:
V = (1/3) × B × H
where B is the area of the base and H is the perpendicular height — the straight-line distance from the apex down to the base plane. The factor of one-third arises from integral calculus: as you stack infinitely thin horizontal cross-sections from the base to the apex, each cross-section shrinks quadratically, and the sum converges to one-third of the bounding prism.
Base Shape Formulas
Square Base
The most common pyramid shape — used in ancient Egyptian monuments — has a square base with side length a:
Base Area = a²V = (1/3) × a² × H
Example: a = 6 m, H = 10 m → V = (1/3) × 36 × 10 = 120 m³. The calculator also computes the slant height s = √(H² + (a/2)²) and the total surface area for square pyramids.
Rectangular Base
A rectangular pyramid has a base with length l and width w:
Base Area = l × wV = (1/3) × l × w × H
Example: l = 8 m, w = 5 m, H = 10 m → V = (1/3) × 40 × 10 ≈ 133.33 m³.
Triangular Base (Tetrahedron)
When the base is a triangle with base length b and base height h_b, the pyramid is called a triangular pyramid or tetrahedron:
Base Area = 0.5 × b × h_bV = (1/3) × 0.5 × b × h_b × H
Example: b = 4 m, h_b = 3 m, H = 9 m → V = (1/3) × 6 × 9 = 18 m³.
Regular Polygon Base
A regular n-sided polygon base with side length a is computed using the cotangent formula:
Base Area = (n × a²) / (4 × tan(π/n))V = (1/3) × Base Area × H
Example: a pentagon base (n = 5) with a = 4 m and H = 8 m gives a base area ≈ 27.53 m², so V ≈ 73.40 m³. Note that a square (n = 4) and triangle (n = 3, equilateral) are special cases of this formula, but those base types use their own dedicated inputs for clarity.
Custom Base Area
If you already know the base area — perhaps from a complex or irregular polygon — enter it directly in the Custom Base Area mode. The calculator then computes V = (1/3) × Area × H without requiring you to decompose the shape.
Slant Height and Surface Area
For square pyramids, this tool also calculates:
- Slant Height (s): the distance from the apex to the midpoint of a base edge, along a lateral face:
s = √(H² + (a/2)²). - Lateral Face Area: the total area of the four triangular faces:
4 × (0.5 × a × s) = 2as. - Total Surface Area: base area plus lateral face area:
a² + 2as.
Unit Conversions
All dimensional inputs — side length, base dimensions, and pyramid height — must be in the same unit. Internally, the calculator converts everything to metres and then re-expresses the volume in m³, cm³, litres, and ft³ for quick cross-reference. Select the unit that matches your measurements: mm, cm, m, km, in, ft, or yd.
Real-World Applications
Pyramid volume calculations are used across a wide range of fields:
- Architecture and construction: estimating the volume of pyramidal roofs, spires, or decorative elements to determine material quantities.
- Civil engineering: calculating earthwork volumes for pyramidal stockpiles of sand, gravel, or coal in mining and logistics.
- Education: verifying results for geometry homework and understanding why the one-third factor appears in volume formulas.
- 3D printing and design: confirming the material volume of pyramid-shaped models to estimate filament or resin consumption.
- Food science: estimating the volume of pyramidal confections or packaging.
Tips for Accurate Results
Always use the perpendicular height H — the height measured at a right angle from the apex directly to the base plane — not the slant height. If you only know the slant height s and the base half-length, you can recover H using H = √(s² − (a/2)²). Ensure all measurements are in the same unit before entering them; mixing metres and centimetres will produce incorrect results.