The volume of the prism is (Length x Width x Height).
To make it easier to work with, let me write (L x W x H).
If you cut each dimension in half, then the volume is
(1/2 x L) x (1/2 x W) x(1/2 x H) .
It's pretty easy to rearrange that, and see what's going on:
1/2 x L x 1/2 x W x1/2 x H =
(1/2 x 1/2 x 1/2) x (L x W x H) =
( 1/8 ) x (original volume) =
( 1 / 2³ ) x (original volume)
Here's the general rule:
If you multiply all the linear dimensions of a drawing OR an object
by some factor 'K' , then
-- all of the linear features, like perimeter, diagonal, circumference,
diameter etc. are all multiplied by 'K' .
-- the 2-dimensional things on it, like area of the drawing, or surfaces
of a prism, sphere or any blob, are all multiplied by K² .
-- the 3-dimensional features, like volume of any prism, pyramid,
sphere, blob, ellipsoid, cylinder etc. are all multiplied by K³ .