Cuboid Perimeters To Volume
For any given cuboid it is possible to measure up to three different perimeters. For example, one perimeter could be measured this way.
Given that cuboid A has perimeters 12, 16, and 20, and cuboid B has perimeters 12, 16, and 24, which cuboid has the greatest volume?
Let the dimensions of a cuboid be $x, y, z$. The perimeters will be $2(x+y), 2(x+z),$ and $2(y+z)$, so dividing each of the perimeters by 2 will give $x+y, x+z,$ and $y+z$ respectively.
The sum of these three terms will give $2x+2y+2z$, leading to $x+y+z$. Thus by subtracting each of $x+y, x+z,$ and $y+z$ we will be able to obtain $z, y,$ and $x$.
Surprising as it may seem, cuboid A has the greater volume.