Lemme see, notice, this is the relationship from side ratios,
to areas and volumes
keep in mind that areas are square figures, involving 2 units,
and volumes are cubic figures, involving 3 units
thus
![\bf \begin{array}{llll} &ratio&relationship\\ lenght&3:4&\cfrac{3}{4}=\cfrac{s}{s}\\\\ area&3:4&\left( \cfrac{3}{4} \right)^2=\cfrac{s^2(area)}{s^2(area)}\\\\ volume&3:4&\left( \cfrac{3}{4} \right)^3=\cfrac{s^3(volume)}{s^3(volume)} \end{array}](https://tex.z-dn.net/?f=%5Cbf%20%5Cbegin%7Barray%7D%7Bllll%7D%0A%26ratio%26relationship%5C%5C%0Alenght%263%3A4%26%5Ccfrac%7B3%7D%7B4%7D%3D%5Ccfrac%7Bs%7D%7Bs%7D%5C%5C%5C%5C%0Aarea%263%3A4%26%5Cleft%28%20%5Ccfrac%7B3%7D%7B4%7D%20%5Cright%29%5E2%3D%5Ccfrac%7Bs%5E2%28area%29%7D%7Bs%5E2%28area%29%7D%5C%5C%5C%5C%0Avolume%263%3A4%26%5Cleft%28%20%5Ccfrac%7B3%7D%7B4%7D%20%5Cright%29%5E3%3D%5Ccfrac%7Bs%5E3%28volume%29%7D%7Bs%5E3%28volume%29%7D%0A%0A%5Cend%7Barray%7D)
what the dickens all that means?
well, you have two cubes, both similar, their ratio, is 3:4,
3 is smaller than 4, thus is from smaller to bigger cube, 3:4 ratio
the smaller cube has a volume, or
![s^3](https://tex.z-dn.net/?f=s%5E3)
of 729 cubic units,
what's the other volume?
well, let us use those proportions above
![\bf \left( \cfrac{3}{4} \right)^3=\cfrac{729}{v}\implies \cfrac{3^3}{4^3}=\cfrac{729}{v}](https://tex.z-dn.net/?f=%5Cbf%20%5Cleft%28%20%5Ccfrac%7B3%7D%7B4%7D%20%5Cright%29%5E3%3D%5Ccfrac%7B729%7D%7Bv%7D%5Cimplies%20%5Ccfrac%7B3%5E3%7D%7B4%5E3%7D%3D%5Ccfrac%7B729%7D%7Bv%7D)
solve for "v"
notice, the numerator in 3:4, has the smaller volume, 729,
the bigger is at the bottom of that proportion