Calculating the Tetrahedral Volumes of Regular Polyhedra

When calculating volumes in units of tetrahedra, what would otherwise be perpendicular measurements are instead measured at an angle of atan(√2) from the horizontal. This angle corresponds to the angle the edges of the regular tetrahedron make with their base. Multiplying the perpendicular value, h, by √6/2 gives h’. See Areas and Volumes in Triangles and Tetrahedra.

A cube of height h measured at 90 degrees from the horizontal, beside a regular tetrahedron of height h' measured at arctan(√2) from the horizontal. h' = h times √6/2.

Polyhedra can be divided up into cones whose bases correspond to the polyhedron’s faces and whose apexes all meet at the polyhedron’s center. This procedure gives the precise volume and, if done correctly, will produce the shorthand formulas that work for all similar polyhedra, as in the following:

Tetrahedral volume a regular tetrahedron = [edge length]³
Tetrahedral volume a cube = 3×[face diagonal]³
Tetrahedral volume a regular octahedron = 4×[edge length]³
Tetrahedral volume a regular rhombic dodecahedron = 6×[long face diagonal]³
Tetrahedral volume a vector equilibrium (VE) = 20×[edge length]³

The procedures by which the volume formulas are established for these and other polyhedra are demonstrated below. The areas of the faces will mostly be given without showing the work. To learn how each is calculated, see Calculating the Triangular Areas of Regular Polygons.

If the cubic volume of a given polyhedron is known, you can also use a conversion factor to quickly find the equivalent volume in tetrahedra. See General Unit Conversions.

General Volume Formula for all Polyhedra

Given the following parameters:

n = number of faces;
n′ = number of alternate faces (if more than one);
r = in-sphere radius connecting center to its perpendicular with the face;
r′ = in-sphere radius connecting center to alternate faces (if more than one);
a = edge length;
face = area of face (in equilateral triangles);
face′ = area of alternate faces (if more than one),

the general formula for the volume of any polyhedron in cubes is

(n/3)×face (in squares)×r (+ (n’/3)×face′ (in squares)×r′…)

and the general formula for the volume of any polyhedron in regular tetrahedra is

n×face (in triangles)×r√6/2 (+ n′×face′ (in triangles)×r′√6/2…)

Examples

Volume of Tetrahedron = a³

n = 4
a = edge length
r = a√6/12
face (in squares) = a²√3/4
face (in equilateral triangles) = a²

Volume (in cubes) = n/3×face×r = 4/3×a²√3/4 × a√6/12 = a³/6√2
Volume (in tetrahedra) = n×face×r√6/2 = 4×a²×(a√6/12×√6/2) = a³

Note: The tetrahedron is more sensibly treated as a single cone with a triangular base, in which case the formula becomes:

Volume (in tetrahedra) = base × height × √6/2 = a² × a√6/3 × √6/2 = a³

Volume of Cube (unit diagonal) = 3d³

n = 6
a = edge length
r = a/2
d = diagonal = a√2
a = d√2/2
r = d√2/4
face (in squares) = a² = (d√2/2)² = d²/2
face (in triangles) = 2d²√3/3

Volume (in cubes) = n×face×r = 6×d²/2×d√2/2 = d³√2/4
Volume (in triangles) = n×face×r√6/2 = 6×2d²√3/3×(d√2/4×√6/2) = 3d³

Note: The cube is more sensibly treated as a cylinder with a square base, in which case the formula becomes:

Volume (in tetrahedra) = 3 × base × height × √6/2 = 3 × 2d²√3/3 × d√2/2 × √6/3 = 3d³

For more information on calculating the tetrahedral volumes of cones and cylinders, see Areas and Volumes in Triangles and Tetrahedra.

Volume of Octahedron = 4a³

n = 8
a = edge length
r = a√6/6
face (in squares) = a²√3/4
face (in triangles) = a²

Volume (in cubes) = n/3×face×r = 8/3×a²√3/4×a√6/6 = a³√2/3
Volume (in tetrahedra) = n×face×r√6/2 = 8×a²×a√6/6×√6/2 = 4a³

Volume of Icosahedron = a³φ² × 5√2 -or- d³/φ × 5√2

φ = golden ratio = (1+√5)/2
n = 20
a = edge length
r = a(√3/12)(3+√5) = aφ²√3/6
d = mid-sphere diameter = a × φ
face (in squares) = a²√3/4
face (in triangles) = a²

Volume (in cubes) = n/3×face×r = 20/3×a²√3/4×a(√3/12)(3+√5) = a³×5(3+√5)/12 = a³×5φ²/6
Volume (in tetrahedra) = n×face×r = 20×a²×(a(√3/12)(3+√5)√6/2) = a³×5(3+√5)√2/2 = a³×5φ²√2

Note: Alternately, the tetrahedral volume of the regular icosahedron may be calculated using the edge-to-edge (mid-sphere) diameter, d, or the long edge of golden rectangle used in its construction: d³ × 5√2/φ. When the edge length = 1, d = φ, and the formula becomes 5(φ+1)√2 ≈ 18.51223

Volume of Vector Equilibrium, VE = 20a³

n = 8
n′ = 6
a = edge length

Volume (in cubes) = (n×tetrahedron)+(n′×half octahedron) = (8a³√2/12)+(6a³√2/6) = a³ 5√2/3
Volume (in tetrahedra) = (n×tetrahedron) + (n′×half octahedron) = (8a³)+(6×2a³) = 20a³

Volume of Rhombic Dodecahedron (unit long diagonal) = 6d³

n = 12
d = long diagonal
a = edge length
d = a2√6/3
a = d√6/4
r = a√6/3 = d√6/4×√6/3 = d/2
face (in squares) = 2×a√6/3×a√3/3 = 2×(d√6/4)√6/3 × (d√6/4)√3/3 = d²√2/2
face (in triangles) = 4×a√6/3×a√3/3×2√3/3 = 4×(d√6/4)√6/3×(d√6/4)√3/3×2√3/3 = d²√6/3

Volume (cubes) = n/3×face×r= 12/3 × d²√2/2 × d/2 = d³2√2
Volume (in tetrahedra) = n×face×r√6/2 = 12×d²√6/3×(d/2)√6/2 = 6d³

Volume of Kelvin Truncated Octahedron = 96a³

Volume of Rhombic Triacontahedron = (dφ)³ × 15√2

For a discussion of the 5-tetra-volume rhombic tricontahedron, see T and E Quanta Modules.

n = 30 (rhomboid faces)
d = length of short diagonal
dφ = length of long diagonal
a = edge length = d√(φ+2)/2
d = 2a/√(φ+2)
r = in-sphere radius = dφ²/2
r = in-sphere radius = a√(1+(2/√5))
face (in squares) = d²φ/2
face (in squares) = 2a²√5/5
face (in equilateral triangles) = d²φ × 2√3/3
face (in equilateral triangles) = 8a²√15/15

Volume (in cubes) = 4a³×√(5+2√5)
Volume (in cubes) = n/3 × face × r = 30/3 × d²φ/2 × dφ²/2 = 5(dφ)³/2
Volume (in tetrahedra) = n × face × r√6/2 = [30] × [d²φ × 2√3/3] × [dφ²/2 × √6/2] = 5(dφ)³ × 3√2

The Geometry of Thinking

Explorations in the energetic-synergetic geometry of Richard Buckminster Fuller