Question

How does a cell ratio of area to volume change because the cell grows larger?

Answer 1

Cell growth causes the area to volume ratio to decrease. This is often because, as a cell grows, the quantity of the cell (its internal contents) increases faster than its surface area (its cell membrane).

Complete answer:
As a cell grows bigger, its internal volume enlarges and therefore the cell wall expands. Unfortunately, the quantity increases sooner than the area , then the relative amount of area available to pass materials to a unit volume of the cell steadily decreases.
In general, small size is important for all cells, whether prokaryotic or eukaryotic. Consider the world and volume of a typical cell. Not all cells are spherical in shape, but most have a tendency to approximate a sphere. The formula for the area of a sphere is 4πr2, while the formula for its volume is 4πr3/3. because the radius of a cell increases, its area increases because of the square of its radius, but its volume increases because of the cube of its radius (much more rapidly).
Therefore, as a cell increases in size, its surface area-to-volume ratio decreases. This same principle would apply if the cell had the form of a cube (below). If the cell grows overlarge, the cell wall won't have sufficient area to support the speed of diffusion required for the increased volume.

Note:
However, increased area can cause problems also. More contact with the environment through the surface of a cell or an organ (relative to its volume) increases loss of water and dissolved substances. High area to volume ratios also present problems of temperature control in unfavorable environments.