A very interesting question...?

Question

If the non-intersecting diagonals of adjacent sides of a cube are at a distance 'd' from each other, find the volume of the cube..
I need proper explanation.....

Answer 1

If 'd' is the distance between nearest endpoints of these no intersection diagonals of the cube, then 'd' = length of the side of the cube.
Hence volume of the cube = (d^3)

But if 'd' is the distance between their midpoints, then thes two midpoints and the midpoint of the coinciding edge for an right angle isoceles triangle with 'd' as its hypoteneuse. In this case,
Assuming the length of a side of the cube = a
(d^2) = {(a/2)^2} + {(a/2)^2} = (a^2) / 2
a = d * (sqrt of 2)
Hence, volume of the cube = a^3 = {d * (sqrt 2)}^3
therefore volume of cube = 2 * (d^3) * (sqrt 2)

And if 'd' is the distance between farthest endpoints of the diagonals, then it forms an right angled triangle along with one of the sides of the cube and one of the adjacent diagonals of the face intersect both these segments.
Now in this case, let us assume that the length of the diagonal is 'a'
and length of the side of the cube is 'b'

as per pythagoras theorom,
(d^2) = (a^2) + (b^2)

but since b is the diagonal of the cube, again
(b^2) = (a^2) + (a^2)
(b^2) = 2 * (a^2)

therefore d^2 = 3 * (a^2)
and a = d / (sqrt 3)

Since the volume of the cube = a^3
your answer would be = d^3 / {(sqrt 3) * 3}

And if, you are picking up random points in those diagonals to measure the distance.... Better go to sleep

Answer 2

This is an interesting problem. At issue is what exactly is the distance between the two non-intersecting diagonals. I say it is the closest distance between these two skew line lines.

This occurs at the corner of the cube where the distance "d" is 1/2 the diagonal, D, of the cube.

Since the side of a cube = sqrt(2)*D/2 = sqrt(2)*d/4,
the vlume in terms of "d" is

V = [sqrt(2)*d/4]^3 = 2sqrt(2)d^3/64 = (sqrt(2)d^3)/32

Answer 3

if i'm reading the question right it's volume = d^3

the non-intersecting diagonals of adjacent sides of a cube would be two diagonals that, for example go from the top left corner to the bottom right corner of each side (were you to look at them face-on).
but if i'm looking at side 1 and side 2 if to the right, then the top of side 1's diagonal is in the top left corner and the top of side 2's diagonal is in the top right corner of the same face.
so if the distance between them is d, then d is the length of a side.
since we're talking about a cube, all the sides have the same length, so the cube's volume is d*d*d = d^3

whew! that would have been so much easier with a picture...

hope it helps

Answer 4

it's simple...the answer is d^3..bcoz in a cube all the diagonals will always hav same length..

Answer 5

area of cube = d x d x d x d
i.e. d cube

Answer 6

whats the problem..........................

volume of cube=1side *2side *3 side
= d *d*d
= d^3