I want to put an 88"H x 15"w x 18" D linen cabinet in my small bathroom which has a 9 foot ceiling. Will I have enough clearance to stand the cabinet up once I get it in the bathroom?
You certainly should have enough clearance. My daughter's bath room has 8 ft ceilings and she had an 81 inch tall linen cabinet installed with out any difficulties.
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The square root of 8068 inches = is just shy of 90 inches.
9 feet (108 inches) less the 90 inches leaves you with 18 inches to spare.
So when the kids whine in math class "when will we ever use this stuff??" you can tell them.
...and using the 15" width in the calculation gives you an 89.3" diagonal. Whichever way you tilt it, it could fit be turned upright even in a room with a 7-1/2' (90") ceiling. If you didn't want to do the math, all you would have needed to do was take a tape measure and measure the diagonal.
This is why I hire all my math done.