Langrange multiplier, confusion at setting up the problem

Question

Question

Find the rectangular box with the largest volume that fits inside the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1, given that it sides are parallel to the axes

Solution

Clearly the box will have the greatest volume if each of its corners touch the ellipse. Let one of the corners (x, y, z) be in the positive octant, then the box has corners (±x, ±y, ±z) and its volume is V = 8xyz.

I don't understand how they volume is equivalent to 8xyz, why isn't $V=xyz$?

Thank you very much

Answer 1

The length of sides are $x-(-x)=2x,y-(-y)=2y$ and $z-(-z)=2z$ so the volume is $$V=2^3xyz=8xyz.$$