Lightroom 4 to Photoshop CS6 confusion
I sized an image in LR4 to these dimension 10.5 x 13.5 ( in the Develop mode).
Images in Lightroom have no particular size, because they are not interpreted according to any particular pixels-per-inch.
They have a certain number of pixels - they have a certain extent on screen when you view them at 1:1 zoom - and LR can tell you the current cropped dimensions of each image; so-many of the originally imported pixels wide, and so-many high, as reduced by your Crop boundary.
The Crop tool can change your pictorial composition (leaving out parts of the picture), and part of that may be changing your picture's shape (aspect ratio). The selection of numbers for width and height in the Crop tool, is purely a convenient way to get to a particular proportion. If you choose 3x2 you will get the exact same result as if you had selected 60x40, 1.5x1, 12x8 and so on.
There are no measurement units on these crop numbers; in particular, they have nothing to do with inches (or centimetres, miles, cubits...) and do not limit or commit the image to be output at any given size.
LR's output size is set by how big a page size and margins you set in the Print module (how big an image cell you provide for it to go into); or by what settings you enter in the Export module.
On the other hand, one of the settings for how an image is to be passed through to Photoshop, is what pixels-per-inch this editing image should get (since even though LR does not care about PPI, technically every image needs to have some kind of a PPI). I think the default is 240, though you can change this.
Then whatever pixel dimensions your current crop shows in LR (or in the Image Size pixel dimensions seen in PS), will get interpreted inside PS using this PPI figure as being the physical printing size of the document. So if the width of your LR crop includes a certain number of original camera pixels (say, 3764) leaving out the rest, this number divided by the PPI entered in the LR external editing settings (say, 240), will be the resulting inches width reported in PS (15.6833 inches).
That is not true the 15.6833 inches stays the same if the PPI is 72, 240 or 300
Once in PS open image size and reads 12.233 x 15.729
The confusion is should they not read the same??
Why the difference?
Photoshop cares about such things; LR simply does not. Each LR image is available for as many different potential output sizes as you want, on web, print, slideshow... since it is output neutral; its output options are limited only by the quantity of original pixel detail available.
In LightRoom an aspect ratio is entered and should be the same aspect ratio as in Photoshop (provided the (constrain proportion) is check)
Take an image in LR crop to 11 x 14 and guess what the image size is not 11 x 14.
While in PS change the dimensions to 11 x 14 and the image shrink to the correct size