A "significant" solid with volume 3 3 π r 3 ?In a painting:there is this formula of volume: V = 3 3 π r 3 It seems to me this is the formula of the volume of some polyhedron inscribed or circumscribed to a sphere of radius r, but I am not expert of the field.I had the task of finding the meaning of the different formulas, that were taken from a web site whose address has been lost, so I need to find the regular solid that corresponds to such formula, if it exists.So I am asking help from somebody more expert then me in the field of geometry of solids, if there is some "significant" solid with the volume given by that formula.

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A &quot;significant&quot; solid with volume             3        3    π      r    3  ?In a painting:there is this formula of volume:  V  =            3        3    π      r    3  It seems to me this is the formula of the volume of some polyhedron inscribed or circumscribed to a sphere of radius r, but I am not expert of the field.I had the task of finding the meaning of the different formulas, that were taken from a web site whose address has been lost, so I need to find the regular solid that corresponds to such formula, if it exists.So I am asking help from somebody more expert then me in the field of geometry of solids, if there is some &quot;significant&quot; solid with the volume given by that formula.

erlent00s · Accepted Answer

Step 1(Converting a comment to an answer, as requested.)The appearance of   π suggests that this is not the formula for the volume of a (flat-sided, straight-edged) polyhedron, whether inscribed or circumscribed about a sphere. Rather, there must be a circular component.Step 2a cone of radius r and heigh       3      r - seems to be the &quot;most natural&quot; one: the target volume decomposes as      1    3    ⋅  π      r    2    ⋅      3      r    =        1    3    ⋅  (  area of base  )  ⋅  (  height  )I&#039;ll note that such a cone arises from revolving an equilateral triangle of side 2r about an axis of symmetry.

A "significant" solid with volume sqrt3/3 pi r^3?

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