Answer
A small surface-area-to-volume ratio will not allow enough materials to pass through the membrane and facilitate the processes a cell needs to survive. However, when the ratio is large, there is a sufficient amount of area for waste to go out of the cell and for nutrients to enter, making life easy.
Work Step by Step
As surface area increases, the volume also increases. For example:
A cube 1 cm all-around has a surface area of 6cm^2 and a volume of 1 cm^3.
This ratio is 6:1
A cube 3 cm all-around has a surface area of 54cm^2 and a volume of 27cm^3.
This ratio is 54:27, or 2:1.
As we can see, the smaller cube is much more efficient at transporting nutrients and waste, while the large cube will struggle with this, due to how much volume it has to cater to.
