Yes. Using a 1by1by1 cube: space diagonal = sqrt(3)
From a right triangle 1-sqrt(2)-sqrt(3)
so i believe the volume of a cube using space diagonals is
V = d^3 / 3sqrt(3) where d is the space diagonal
can anyone help me prove this? the question this relates to is actually the volume of a rectangular prism in terms of space diagonal, but to understand the cube scenario would be a good starter.
apologies for not using latex, i havent used it in a while and need to re-educate myself
thankyou in advance
. . . Right!
We have a cube with side
Code:x *--------C / /| / / | x *-------* | | | *B x | | / | |/ x A*-------* x
Consider the "floor" of the cube.
Code:*-------* B | * | | * | x | * | A *-------* x
We see that: .
Now "slice" the cube through vertices and
We have this cross-section:
Code:* C d * | * | x * | A *-----------* B _ x√2
And we have: .
The volume of the cube is