The dimension in pixels (Image Size) is the important detail for using any image. Around 300 pixels per inch is a reasonable and optimum and proper printing goal for photographs. 200 dpi can sometimes be acceptable printing quality, but more than 300 dpi is not of much use to printers (for color photos). Many 1-hour photo lab digital machines are usually set to print at up to 250 pixels per inch. 250 to 300 dpi is a reasonable and optimum printing resolution for photos. FWIW, I'm old school, and I learned the term for printing resolution as "dpi", so that's second nature, simply the name of it. Some do call it ppi now, same thing, pixels per inch. Printers do have their own other thing about ink drops per inch (which is about quality of printing, not about image resolution), but here, we're speaking about printing resolution of image pixels.
There are two situations when printing images, print shops or home printing.
Planning ahead to provide enough pixels so that the result will be 250 to 300 pixels per inch is a very good plan. If your image is too large, the photo shop will first resample it to this size. If too small, it simply won't print as well.
However, most photo editors will also provide an option to "Scale to fit media" or "Best fit to page", which will scale the image to fit the specified paper size (similar to A above). This can be borderless if so specified in the printer Properties. This will print at a new dpi which will fit the paper size.
When the print and paper shapes differ, option A print shop "fit" typically fills all of the paper, cropping one dimension to leave no white border in the other. Option B photo editor "fit" typically is the opposite, not cropping at all, but leaving thin white space in one dimension.
Either way, it is good if your plan properly prepares the image for printing. Sufficient pixels is important, but first cropping the image so that the image SHAPE actually matches the selected paper SHAPE is also a very important concern. Different paper sizes are different shape. And we need to provide the necessary pixels. The most-minimal simple calculation for that acceptable image size for printing is:
(The actual dpi calculator is below). This little calculator will serve two purposes:
But this dpi number does NOT need to be exact, 10% or 15% variation won't have great effect. But planning size to have sufficient pixels to be somewhere near the size ballpark of 240 to 300 pixels per inch is a very good thing for printing.
Long dimension fitted
Short dimension fitted
Preparing the image shape to fit the paper shape is necessary, because paper and image are often different shapes.
Aspect Ratio is the "shape" of the image - the simple ratio of the images long side to its short side, maybe long and thin, or short and wide. And every paper size seems to be a different shape too. Shape and size are two different properties. To print an image, we can always enlarge the Size, but the image shape needs to match the paper shape (which is done by cropping). If this Aspect Ratio subject is new, see Image Resize - Aspect.
The calculator below specifies just one paper dimension, either the short or long one. That dimension is fitted to the paper, and the other dimension will float (will vary, to be whatever the image shape actually is). Ideally, you will have already cropped the image shape to match the selected paper shape properly, to be your choice. Otherwise, you may get a surprise about what image area has been cropped off (if it won't fit the paper shape).
Repeating, the important thing about cropping to fit the paper is that you will like the results much better if you first choose the cropping yourself. If you wait for the machine to simply cut off the sides automatically, then maybe that's not so pleasing (not your preferred choice of what to include).
The idea is to first crop the image to be the same shape as the selected paper.
See Image Resize about how to plan this necessary resize and/or crop.
That procedure is, FIRST crop image to paper SHAPE, and then resample to fit paper SIZE.
Otherwise, printing will just cut off the part of the image that won't fit the paper.
The calculator below asks if your print should be fitted to the Long or the Short image dimension. For example, if printing 8x10, specify 8 short or 10 long (which way to fit it). Or, if you have already properly cropped your image to match paper shape, then it doesn't matter which you choose, both long and short results will match the paper shape.
First, fitting an Image to the Paper Size - In the calculator below, specify either the Short or Long Dimension of the paper. This is the important dimension you want to be exactly filled (fitted to the paper). The other dimension will float, possibly to be smaller or larger than the paper, determined by their shapes (aspect ratio). Normally, a copy of the image should have previously been cropped to be the same aspect ratio as the selected paper shape.
When a scanner scans at 300 dpi, it creates 300 pixels per inch of dimension scanned. Scanning 10 inches at 300 dpi creates a 3000 pixel image.
For scanning, we can calculate the exact numbers required. And we are able to scan at any numbers like say 1347 dpi, or at 314% scale, and that works well enough, but purists consider it good to scan a little larger, specifically at one of the scanner default menu settings, like 150, 300, 600, 1200, 2400, 4800 dpi values offered (assuming 100% scale). Specifically, the next offered step larger than your desired size. Then resample smaller to the desired size. The reason is that the scanner hardware (the sensor pixels and the carriage stepping motor), can only do those specific offered values. Anything else is approximations and resampling. It is a small hard-to-see difference (maybe a geeky thing), however the photo editor can do this resample better, after it has all the data. Crop FIRST as desired to improve composition, and then resample the final version to smaller desired size to print.
When a printer prints at 300 dpi, it spaces the pixels onto paper at 300 pixels per inch of paper. Printing 3000 pixels at 300 dpi prints a 10 inch image on paper.
The straight-forward way to scale for printing is to simply compute "pixels per inch" for the inches scanned, and then recompute those pixels over the inches printed (called scaling, as mentioned in the scanning Results). The scanner will have its Input and Output dimensions to show this. Also we have photo editor tools to make this resize be easy. See Image Resize and Scaling.
However a shortcut for the same scaling concept is this:
The ratio of (scanning resolution / printing resolution) is the enlargement factor.
For example (in general - speaking of any size original):
Scan at 600 dpi, print at 300 dpi, for 600/300 = 2X size (to print double size or 200% size)
Scan at 300 dpi, print at 300 dpi, for 300/300 = 1X size (to print original size or 100% size)
Scan at 150 dpi, print at 300 dpi, for 150/300 = 1/2X size (to print half size or 50% size)
Or scan small film at 2700 dpi, print at 300 dpi, for 2700/300 = 9X size. If from full frame 35 mm film (roughly 0.92 x 1.41 inches), then 9X is about 8x12 inches (near A4 size). Film is typically small, requiring more scan resolution for more pixels for more print enlargement. The reason to scan at high resolution is for "enlargement", specifically to create enough pixels to print a larger print at about 300 pixels per inch. Scanning larger than any reasonable future use is likely pointless.
This is a concept called "scaling", and this enlargement concept is true for scanning anything, photo prints, documents, film, etc.
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