What size images do I need for common print sizes, such as wallet size, 8x10, and 16x20? Is there a general formula?
How could that size change if, for example, a 16x20 was printed on canvas rather than photopaper?
There are some general rules you can use to determine the "maximum" (I use that term loosely) print size. Keep in mind that the quality of a print is often more dependent on what is being printed than its size in megapixels, and even if your image size is not dense enough to mathematically fit onto a certain page size, you can still blow most images up pretty large without significant or noticable loss in quality.
Anyway, as a general formula:
Screen/camera pixels do not directly translate to print pixels (PPI, pixels per inch), so you need to translate your image pixel size into a print size by dividing your screen pixel sizes by the PPI of your print. Assuming you have an 8mp camera, your image sizes are likely to be around 3200 x 2400 or so. If you print at the native "photo quality" PPI of an Epson printer of 300ppi, you would end up with the following:
That boils down to an 8.5 x 11 print at full resolution without any loss of quality or resolution in the translation between screen and print. However, this is not a general rule for print quality...it is simply a rule for ascertaining the lossless print size for a given image size. You could still blow up your 8mp photo and print it at 11x16, or 13x19, and still have a good print.
You can use this formula in reverse as well, to determine the screen pixel size your image would need to be to be for a particular paper size:
Using this, you could determine what size image you would need for a 2x3 inch print:
There are other things you can do with your image to improve its quality when printed. Sharpening an image before printing, assuming you do so with the proper care, can improve the quality of your printed copies. It is also possible to use image editing software, such as Photoshop or QImage, to digitally blow up your image to a higher resolution, perform some post process sharpening, and print at even larger sizes.
Beyond this relatively simple explanation, the discussion of how to translate a photo image to its printed paper size is a relatively complex topic. It is possible to print at a variety of PPI, from as low as 100 to as high as 480 or higher. The PPI you choose to print at, the size of your image, and the quality of the algorithms you may have used to scale your image up or down, the texture, brightness, thickness, and color of your paper, and the kind of printer all determine how good a photo looks when printed.
The information above is quite good, so I won't try to compete, but here is a nice infographic:
The boxes are the number of megapixels for a print of the size in inches according to the scales on the axes. This is at 300ppi, which is a standard for the print resolution of many images.
This great graphic comes from an article at D215.
jrista has the start of the formula, and it covers images viewed at arm's length quite well. But that 'conventional wisdom' devolves into unreasonable numbers as soon as you get to anything "big", say even a 16x20... requiring 5-6000 px. And if you hit poster size, say 30x40... 9000x12000... 108 MPix?!
When you're talking about really big prints, it's important to realize that the vast majority of folks want to view the entire image, they're going to stand back far enough that they can see the whole thing comfortably. At that distance, the human eye can't resolve 300ppi density. So you can easily decrease the density when you start getting bigger and bigger. You can find lots of discusion about the fidelity of the human eye and it's relation to viewing distance and quality of images if you google for "apple retina display arc second", in particular, this article is a great intro to the concept.
Many print vendors use this same reasoning to set their minimum print resolutions for larger print sizes, at SmugMug our minimum values take this into account. (disclaimer in case it wasn't obvious, I work for SmugMug.) That's not saying you should print at those sizes, just saying that we won't let you print anything smaller.
As an example, I have a 30" wide shot that's printed at a mere 100 ppi density. (roughly, the file is like 3123px wide or so.) Being 30" wide, folks don't hold it at arms distance to view it... they tend to stand 3 or 4 feet away and look at it on the wall. No one comments on the resolution. The exception is a couple of photographers that have seen it... and then it's only after they've stepped back to enjoy the scene that they step up close, like a couple inches away, and look at the image in a few key places to spot check the focus. On the bell curve of viewers, those folks are way way way outside the norm.
Most printers print in 300 dpi.
If you use Photoshop you can go to Image --> "Image Size" --> uncheck "Resample Image" and then change the Resolution to 300 dpi (pixels/inch).
Then the you will see maximum Width and Height in cm (or inches) that you can print this image without resizing it.
If you want to print on canvas then you will probably use the same size as for photopaper (if the canvas printer will print in 300 dpi) but for canvas print you probably will need to sharpen the image more than for ordinary photopaper.
If you want to print 8x10 at 300 dpi then you will probably need an image that is 2400 pixels (width) x 3000 pixels (height).
For 16x20 at 300 you will need image that is 4800 pixels (width) x 6000 pixels (height).