There’s a very simple relationship between the distance to the screen and the size of the image. Basically, the distance divided by the half the size is always equal to a constant.
However, the value of that constant depends on what kind of projector (and especially what kind of lens) you’re using. Think of it like the zoom on a camera. The only difference here is that in order to change your “zoom”, you (probably) have to switch lenses.
So, if you want to have a short throw distance (the distance from the projector to the screen), you just need a very wide angle lens. Keep in mind that you have to keep things from getting too ridiculous: if you want the projector to be 3 feet from the screen there’s no way you’re going to get a 9 foot image.
Basically, you have three elements that are all related: Distance from the screen, size of the image, and what kind of lens you’re using in your projector. You can pick two of them and that will determine what the third will be. All you need to figure it out is what’s called a sectional drawing of your theater; basically a side view where you can draw out the angles.
Instead of trying to put the projector really close to the screen, I’d be more likely to put it farther away, and higher, like above the audience’s heads. If the projection is coming in at a steep angle, there won’t be as many shadows cast on the screen. The projection basically shoots over the heads of the actors and hits the screen. Again, you have to have some sense of what’s possible; there will be a zone about 3–5 feet deep directly in front of the screen, whose size will depend on how steep of an angle you’re using, where the shadows of actors’ heads will show up on the screen, but it’s not that big of a problem.
What can be problematic with a steep projection angle is the problem of keystoning. Basically, the top of the image is closer to the projector than the bottom of the image, so it’s smaller, and the image looks like an upside down version of this:
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Fortunately, many projectors have a function, called keystoning, where they digitally compensate for this by making one side of the image smaller than the other, which cancels out the keystone effect.
The other way to go is rear projection, in which the projector is located behind the screen. There are disadvantages to this as well. I don’t know how your stage is set up, but you very well may not have a lot of room behind your screen for the projection to spread out, so you may need a really wide lens here too. The other problem with rear projection is that because the light is passing through the screen, it’s not as bright. They make special rear-projection screens that minimize this, but they’re expensive. A shower curtain is actually a decent compromise between transparency and cost.
If you don’t need to be projecting video, then it’s possible to do it with non-digital technology as well. This might be cheaper, and your school might have some equipment laying around somewhere unused. Slide projectors are bright and have many different size lenses; overhead projectors are tacky, but if they fit with the theme of the show, they could be perfect. They’re very bright, and have a really wide beam angle. My high school did a production of Ubu Roi a number of years ago, and the basic idea of the whole play was that it was all potty humor. So we used drawings made with overhead pens as a projected backdrop. We rear-projected it, and had two overheads side by side, whith marks on them to line up the two images. The projectors were on a power strip so that they could be switched on and off together.
Congratulations if you’ve taken the time to read all of this; I tend to get long-winded about these things, and I apologize, but I hope I’ve provided you with some useful information.
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