Under Crop in the Inspector will be the area selection dimensions
Adobe Photoshop (14.0):
File > Open
The Import PDF dialog will open. If there are multiple pages, select the page(s) you wish to open and click OK.
Window > Info (if not already shown)
Select the Marquee tool
Alternatively you can use the Crop tool
Select the area of the PDF you want to measure
The dimensions will show in the Info dialog box.
If they are shown in pixels, go to Photoshop > Preferences > Units & Rulers
Change Rulers to Inches, or the desired unit and click OK.
Alternatively you can hit ⌘R to show Rulers, right click on the Ruler and a menu will show where you can change the units.
Adobe Illustrator (14.0):
File > Open
If there are multiple pages, select the page you wish to open and click OK.
Window > Info (if not already shown)
In most cases you can use the Selection Tool (V) to select the object you wish to measure. In some cases there may be clipping masks and you'll need to use the Direct Selection Tool (A) to select a specific object. In this case the PDF has a cut line around the sticker and that is what we'll select.
The object dimensions will show in the Info dialog.
You can also use Window > Transform to view (and change) dimensions
To see them in different unit measurements, go to Illustrator > Preferences > Units and change the General Units drop down. You can also hit ⌘R to show Rulers, then right click on a ruler and a menu will show where you can change the units.