This article is about relation between digital image size in pixels and photographic print size.
First calculator in this article recommends photographic print size for given digital image dimensions in pixels.
Let's define the problem:
We have digital image with known size in pixels, for example, 3264 x 2448 pixels, and the set of standard photo print sizes used by photographic printing services. Photo print size name defines linear dimensions of photo print. For example, 4''x6'' or 4R photo print size means that photo print has 102x152 millimeters dimensions.
We need to choose maximum photo print size which allows us to print digital image without quality loss.
I've created the handbook Standard photographic print sizes for defining standard photo print sizes, which can be edited to provide missed sizes.
The only special knowledge we needed to solve the problem is about quality. It can be easily found on the web. Photographic quality (at an arm's length viewing distance) requires that print resolution should be not less than 300 DPI (dots per inch) or, the same, 300 PPI (pixels per inch). Still acceptable quality requires that print resolution should be not less than 150 DPI. Everything else is simple math.
Look at the picture below
Each photo print size is converted to pixels, assuming that 1 inch holds 300 (150) pixels. The obtained size in pixels (taking into account aspect ratio, more on this below) is compared to digital image size. If print size in pixels is greater than digital image size (see picture, print size on the right), then it does not fit, since we have to enlarge the image and get resolution worse than 300 DPI. If print size in pixels is less than digital image size (see picture, print size on the left), then it does fit, since we have to shrink the image and get resolution better than 300 DPI.
The calculator chooses print size with maximum linear dimension, which does still fit. (Smaller size won't be a problem since we can print with resolution up to 1200 DPI.)
Second calculator in this article finds resulting pixels per inch value for printed image and how many pixels were cropped during scaling. Let's define the problem.
We have photo print with known dimensions in centimeters, printed from digital image with known dimensions in pixels. Usually, the aspect ratio of printed image is not the same as aspect ratio of digital image. The image is scaled during printing, but its aspect ratio remains constant. This leads to unwanted effects.
Look at the picture below
We have two ways to scale:
first - scale with cropping out the part of image
second - scale without cropping, but with empty spaces on photo print.
Since second looks ugly, I used the first.
Thus, we have to find resulting image resolution in DPI, and how many pixels were cropped due to difference in aspect ratio. First one is easy - the dimension in pixels (width or height) which is not cropped is divided by corresponding photo print dimension in inches. The second one is difference between used pixels on cropped dimension and original digital image pixels.