The most accurate way to quantify microbes on a petri dish is to dilute them to the point that you expect about less than 100 microbes. Then you let them grow and count them taking into account your dilution. Each little dot you see will have about 100 million to a billion bacteria (assuming you are plating bacteria).

Unfortunately, if you don’t dilute the microbes somehow, you’ll get “lawns” which are smears of cells. There is really no good way to count the number of cells in a lawn. Using the area covered by the cells is not that accurate because cells will grow until they get to close to other cells which gives you a very inhomogeneous result of bacterial growth. Also, it’s possible that some parts of the dish may be more suitable for growth than other parts which would give you clusters of cells in certain parts. The height of the microbes is irrelevant because they will generally only grow on the top of the dish in a very thin section that you would have trouble measuring.

If you are starting with a dish with microbes, then you could scrape off the top and put them into a liquid that won’t kill the microbe, but it won’t grow too much in (phosphate buffered saline is usually best). Then make sure the solution is well mixed. You could then use a pipet to transfer a small amount of the liquid to a large amount of liquid to dilute it, and plate the result. Let the plate incubate overnight (best is around 100 Fahrenheit). Keep diluting until you get a dish with circular, well separated bacterial colonies that you can count. Take into account the dilution used to get those well separates bacteria and then you can have an idea of the total amount of bacteria on the original plate. This would be a lot more work intensive, but also a lot more accurate than using area of microbes.

While you could use a ruler, I doubt you would get any precise measurements, so I'm not sure how you could measure the volume of the bacterial growth. Depending on the bacteria, you may be able to detect metabolism by the dissolution of the agar, if you have a very fine scale to weigh them on (you would need that, though).

In our lab, we measure microbial growth using a spectrophotometer to estimate the density of cells in solution. The more turbid the solution gets, the more microbial growth we have. I understand that you probably don’t have this instrument in your classroom, and I imagine it’s probably pretty difficult to measure how many cells are on a petri dish. Can you see individual colonies, or do the bacterial form into big globs? If you can’t see individual colonies, it’s hard to arrive at a quantitative result of bacterial growth. But, you can still use a dilution method and petri dishes to arrive at a useful estimation! You can do this with a colony forming unit assay, which is used to estimate the number of bacteria cells in a sample.

For this assay, a solution of bacteria at an unknown concentration is serially diluted in order to obtain a plate with a countable number of bacteria (spots of bacteria colonies). The standard formula to obtain a colony count is CFU = total dilution used to make the plate x the volume of cell solution added to the plate. When I’ve done this assay, I’ve often had to dilute it by 1,000,000,000 just to get a countable number!

A useful website explaining this protocol can be found: here