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I am designing a flyer for the screen and the client insists the logo should be 71mm exactly. Does the resolution plays any role on the viewer, some may see it bigger, some smaller? Thank you.

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Millimeters, inches, centimeters, picas.. all do not translate universally to pixel sizes.

Pixels are not a physical object, there's no measured size for a pixel.

Physical measurements only relate to printed materials, never anything on screen.

You need to ask your client if a pixel size is acceptable. Perhaps show him/her what 71mm would be at 72ppi (201px), 150ppi (419px), 300ppi (840px), because it will be different based upon the resolution of the image.

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    To be more accurate a pixel has a well defined size, it is nothing. It has no size its a points sample. Pixels have a pitch. If a image intended for digital consumption than saying its is 71 mm wide for example makes no sense. Since the pitch is different on each device. – joojaa Mar 26 '18 at 17:34
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    +1. The one thing that might make this "slightly untrue" is if the spec calls for a PDF final, in which case a target physical size is sort of implicit. The OP might in that case assume ~120px per cm (the 840px) – Yorik Mar 26 '18 at 18:09
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    The pixels my screen is made of are real physical objects... But you can't necessarily control how your pixels will translate to screen pixels or how big a screen pixel is in millimeters. – user253751 Mar 26 '18 at 22:52
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    @Scott With a cathode-ray tube (CRT) monitor, that’d be an accurate-enough description, but the overwhelming majority of monitors these days instead are based on light-emitting diodes (LED), and for that matter, rarely have a glass covering. And in an LED monitor, there is an array of LEDs for each pixel (specifically, there is a diode for each red, green, and blue sub-pixel, and a group of them forms each pixel). You can even see them if you look closely (easier for lower-resolution screens). So yes, in an LED monitor, pixels are physical objects, and the monitor is made of them. – KRyan Mar 27 '18 at 3:40
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    (Intentionally glossing over various sub-pixel geometries and how sub-pixels are sometimes shared between adjacent pixels because it can produce higher resolution/better colors.) – KRyan Mar 27 '18 at 3:41
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The size of any image presented on a computer screen will change with the resolution settings of that particular computer. There is no assurance that an image on your machine will be the same size as that image on my machine.

Additionally, the software being used to display the image will create variations in the displayed size. Web browsers can be adjusted to different zoom levels, image editors always have zoom capability, even PDF readers allow adjustable zoom.

Your client should provide specific parameters relevant to the above considerations to enable you to accomplish his objective.

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