Generally speaking, there is no set number of paragraphs for any length of research paper, or any other type of writing, for that matter, nor is there a set number of sentences per paragraphs. For a paper of the length you're describing, you would plan on an introduction and conclusion that might be a third to half a page long, and probably seven to ten paragraphs in between. The length of your work will be affected by the topic, the type and number of direct quotes you include, method of citation employed, and, as elementary as it sounds, your choice of vocabulary, because more sophisticated language, particularly specialized vocabulary, will likely include longer terminology.
That being said, I've found that when an instructor (including myself) gives a desired length for an assignment, it is meant (or should be) as more of a guideline than a set-in-stone requirement, intended to give the student a general idea of the depth of information the instructor expects to see presented. By saying a paper should be five pages, the instructor hopes to stave off the underachiever who turns in a most superficial examination of the topic in, say, two pages, as well as the overachiever who might be tempted, in the name of earning a good grade, to write something in the tradition of the doctoral thesis.
By the same token, the number of paragraphs you end up with will vary according to your topic and writing style. A paper of this length will likely have paragraphs that range from five to eight or nine sentences; fewer than five sentences in a paragraph from the paper's body might be an indicator that more detail is needed, while more than nine or ten sentences might indicate that too much information is being attempted in one paragraph and the writer needs to do some editing or breaking of the paragraph into two or more.
I would advise accumulating whatever research you'll be using and write out a first draft with little or no thought to polishing just yet--simply get everything on paper, forming your paragraphs without too much angst, so that you have a starting place. From there, you can sit back and evaluate if you are in the five page area in terms of length; you can ascertain where there might be holes in your research or ideas, requiring more information be added, or where you perhaps have an overabundance of material, which may cause you to either cut some of the extraneous material, or possibly refine or adjust your original topic to account for the proliferation of information.
There's no set number of paragraphs for a 5 page essay but when we had our 5 page paper due we had about 11 paragraphs with 2 or 3 relating to parts of our essay. When you feel you are finishing a point you are allowed to create a new paragraph and make sure you do not stretch it. If you are still on one point, but it is taking up the whole page feel free to split it apart.
Assuming that each paragraph is 5 sentences long (standard length paragraph) and there are 5 paragraphs to a page, there should be 25 paragraphs to a 5 page paper.
A few things to keep in mind are: 1) that sentence length and paragraph length can vary greatly, which will effect the number of paragraphs needed to fill 5 pages, 2) that the first page and last page of your paper is going to have larger paragraphs (for your introductory and closing statements)then the body of the paper, and 3) that clearly communicating information and your ideas is much more important then the number of paragraphs you use.
I hope this helps!