Some demonstrations of volume formulas
Volume of a pyramid
is 1/3 base times height
There are 3 kinds of the
smaller tetrahedral blocks
(4 if you regard the left double corner as different from the right)
They all have the same volume.
The Platonic tetrahedral block
has twice the volume of
each of the other blocks.
Knowing all the volumes of the four smaller pieces are the same
and that the larger yellow block is twice the volume of his little brothers,
it is easy to figure the volume of the various figures assembled with the blocks.
Some examples:
A cube can be formed from a yellow tetrahedron block and 4 octanes.
Since the octane has half the volume of the tet you know the
volume of the cube thus constructed equals
1 tet + 4*(tet/2) =1 tet + 2 tets = 3 tets.
You can construct an octahedron from 8 octanes
So you know this octahedron has the same volume as 4 tet blocks.
Kirby Urner has a page where he has set the tetrahedron volume as 1.
It's worth a look:
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