Question

How does surface area to volume ratio relate to cell division?

Answer 1

The biological basis for the growth of an organism is cell division. This process allows for the development of a daughter organism in simple unicellular organisms such as yeast (Saccharomyces). Cell division allows the creation of an individual from the one-celled zygote process to a fully functioning adult with several trillions of cells in complex organisms such as humans.

Complete answer:
Cells need to produce chemical energy (via metabolism) to survive and this requires the exchange of materials with the environment. The rate of metabolism of a cell is a function of its mass/volume (larger cells need more energy to sustain essential functions)
The rate of material exchange is a function of its surface area (large membrane surface equates to more material movement) As a cell grows, volume (${units^3}$) increases faster than the surface area (${units^2}$), leading to a decreased SA: Vol ratio. If the metabolic rate exceeds the rate of exchange of vital materials and wastes (low SA: Vol ratio), the cell will eventually die. Hence growing cells tend to divide and remain small to maintain a high SA: Vol ratio suitable for survival.
If you are a cell that relies on diffusion through your cell wall to acquire oxygen, water, and food and get rid of carbon dioxide and waste materials, this is important. The outside is unable to keep up with the needs of the inside as you get larger. With agar cubes that have been soaked in NaOH solution, we can see this. The' nutrients' have spread to the middle of the smallest cube, but the largest cube is mostly "starved" in the center. Your SA: V has become so small that your surface area is not big enough to provide your insides with nutrients if you are a cell-like the largest cube.
You've got a tonne of options. You can:
Dying
Adjust the form too long and narrow or thin and flat
Separate into smaller cells

Note:
The SA/V ratio still increases as the cell expands, but as the cell starts to split, that ratio begins to shrink, and the SA/V ratio is cut in half by the time the cell is ready to go through cytokinesis, so the cell divides to produce two new cells.