This is determined by the pixel density. The pixel density, most of the times expressed in DPI or PPI (dots/pixels per inch), is a property of the image.
If you have an image of 900 x 600 pixels, the size at which it will be printed depends on the DPI of the image. For example, if it is set to be 300, then the image will be printed at 3 x 2 inches.
So to know how large your image will be printed in centimeters, you have to divide both width and height by the DPI value and then multiply both values with 2.54 to convert the inches to centimeters.
The other way around also works. An A3 sheet of paper is 29.7 x 42.0 centimeters. Divide by 2.54 to get the size in inches: 11.69 x 16.54. Then multiply with the DPI setting you want to get the size in pixels. With a DPI of 300 this gives you 3,507 x 4962 pixels.
Update about printers
As far as I know it wont influence the size (as long as the printer support that size of paper), but it can influence the quality somewhat. An inktjet printer has a maximum DPI. For example, an HP Deskjet 2050A has a maximum DPI of 4800 x 1200. When your image has a lower DPI (very likely :-)), the printer has to somehow "resample" the DPI. In such a case it is best that both the horizontal and the vertical maximum DPI of your printer are divisible by the DPI of your image. If you set the DPI of your image to 800, you might get some quality loss because 1,200 / 800 is not a round number. Setting it to 300, 150, 100 will probably yield better results because both 4800 and 1200 are divisible by those values.