A regular octahedron can be augmented on 3 faces to create a triangular frustum
A frustum's axis is that of the original cone or pyramid. A frustum is circular if it has circular bases; it is right if the axis is perpendicular to both bases, and oblique otherwise.
The height of a frustum is the perpendicular distance between the planes of the two bases.
Cones and pyramids can be viewed as degenerate cases of frusta, where one of the cutting planes passes through the apex (so that the corresponding base reduces to a point). The pyramidal frusta are a subclass of the prismatoids.
Two frusta joined at their bases make a bifrustum.
where a and b are the base and top side lengths of the truncated pyramid, and h is the height.
The Egyptians knew the correct formula for obtaining the volume of a truncated square pyramid, but no proof of this equation is given in the Moscow papyrus.
The volume of a conical or pyramidal frustum is the volume of the solid before slicing the apex off, minus the volume of the apex:
where B1 is the area of one base, B2 is the area of the other base, and h1, h2 are the perpendicular heights from the apex to the planes of the two bases.
the formula for the volume can be expressed as a product of this proportionality ?/3 and a difference of cubes of heights h1 and h2 only.
By factoring the difference of two cubes ( a3 - b3 = (a-b)(a2 + ab + b2) ) one gets h1-h2 = h, the height of the frustum, and ?(h12 + h1h2 + h22)/3.
Distributing ? and substituting from its definition, the Heronian mean of areas B1 and B2 is obtained. The alternative formula is therefore
1.^ The term "frustum" comes from Latinfrustum meaning "piece" or "crumb". The English word is often misspelled as frustrum, a different Latin word cognate to the English word "frustrate". The confusion between these two words is very old: a warning about them can be found in the Appendix Probi, and the works of Plautus include a pun on them.
^William F. Kern, James R. Bland, Solid Mensuration with proofs, 1938, p. 67
^Nahin, Paul. An Imaginary Tale: The story of . Princeton University Press. 1998