How do you calculate the surface area-to-volume ratio of a cell?
You could consider the cell to be a sphere and them just calculate it
if you know the radius caluculate it like this
with r the radius of the cell
The next question: How would you measure this radius in the first place?
Most cells are spherical in suspension. That is, when they are freely suspended in a liquid medium, they exert the same forces in all directions, thus making them spherical.
You can take a picture in a camera equipped microscope at a known magnification and use a scale bar to measure cell radius. There are also methods to automate this through image processing
Note: This rule however does not apply to plant cells (rigid cell wall), RBCs (flattened) or many bacterial cells that retain a different shape. In these cases, you can approximate the cell to be a cylinder, disc, cuboid, etc and use known formulae, or if you have access to a confocal microscope, you can get 'slices' very much like a CT scan, and you can build a 3D model of the cell from it. Calculation of surface area and volume shouldn't be difficult after this.