In Fuller’s geometry, areas and volumes are measured in unit triangles and tetrahedron, rather than unit squares and cubes (see Areas and Volumes in Triangles and Tetrahedra). Fuller seems to have made unit conversions between his 60° coordinate system and the conventional 90° coordinate system unnecessarily complicated with the synergetics power constants (see Pi and the Synergetics Constants). There may be more to his constants than I’ve been able to untangle, but the following two formulas for areas and volumes, which are simply the inverses of the conventionally-calculated area of the unit triangle and unit tetrahedra, are more straight-forward; and they work, at least in two and three dimensions.
The area, in unit squares, of a triangle is conventionally calculated as the base times the height divided by two. An equilateral triangle of unit edge length would therefore have an area, in unit squares, of 1 times √3/2 divided by 2, or √3/4. To convert any given area in squares to its equivalent area in equilateral triangles, we would simply multiply by the inverse, 4/√3, or 4√3/3.
area in squares × (4√3/3) = area in equilateral triangles
For example, the area of an equilateral triangle of edge length 4 is conventionally calculated as (4×2√3)/2, an irrational number approximately equal to 6.9282. Multiplying this number by 4√3/3 returns exactly 16, or 4×4, or 4².
The volume, in unit cubes, of a tetrahedron is conventionally calculated as the base times the height divided by three. A regular tetrahedron of unit edge length would therefore have a volume, in unit cubes, of √3/4 × √6/3 ÷ 3, or √2/12. Therefore, to convert any given volume in cubes to its equivalent volume in in tetrahedra, we would multiply by the inverse, 12/√2, or 6√2.
cubic volume × (6√2) = volume in tetrahedra
For example, the cubic volume of the unit-diagonal cube is (√2/2)3, or √2/4 which when multiplied by (6√2) equals 3 tetrahedra. A rhombic dodecahedron with its long diagonal of unit length has an edge length of √6/4. The formula for its cubic volume is (16√3)/9 × edge length3 = (16√3)/9 × (√6/4)³ = 16√3/9 × 6√6/(16×4) = 18√2/36 = √2/2, which, multiplied by 6√2 = 6 tetrahedra.
The Geometry of Thinking 