Volume of each of the 4 or 6 square pyramids that fit in a cube is 1/3 of the cube?

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Websites show different arrangements of a cube with square pyramids - 3 or 6:

  • 3 square pyramids fit in a cube
  • 6 square pyramids fit in a cube

Volume of square pyramid = (1/3) (B) (h)

Do each of the 6 square pyramids have a volume of 1/3 of the cube (when 6 square pyramids fit in a cube)?

Do the 6 square pyramids become 3 equal pyramids, each with 1/3 cube volume?

UPDATE: Image link: https://i.ytimg.com/vi/5pBigy5Cwo8/maxresdefault.jpg

Video link: https://www.youtube.com/watch?v=5pBigy5Cwo8

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Yes, the correct formula for the volume of a square pyramid is $V=Bh/3$. However, it is up to you to decide what $B$ and $h$ choices are relevant to the problem at hand and how to apply the formula to get a reasonable answer.


The two different answers you are seeing (3 versus 6) depends on where each geometer decided to put the peaks of their pyramids.

If one puts the "peak" in the very center of a cube that has side lengths $s$, then each face of the cube creates its own identical pyramid (1-of-6) with $V=Bh/3$ but with $B=s\cdot s$ and $h=s/2$.

On the other hand, if the geometer decided that the pyramids didn't have to be identical, they might instead put the peak on an outside corner. In which case, each of the three faces touching that corner do not create pyramids as they would have zero height, but each of the three faces opposite that corner would create pyramids (with $V=Bh/3$ but $B=s\cdot s$ and $h=s$).

Either way you get $s^3$ for the final volume of the cube. The 3-or-6 distinction just depends on how you precisely decide to cut it up. (Heck, we could break the original cube into 8 smaller cubes and then say there are 24-or-48 pyramids to work with... or cut it into 27 smaller cubes... etc.)