I have a different answer from other folks: pixel is the correct unit for areas, and you do need dimensional analysis. The discrepancy is that "3840 pixels wide" is not a strictly correct measurement. Instead, "pixel" is a natural-language abbreviation for different units at different times, and it takes some context and judgment to expand the abbreviation appropriately.
The unabbreviated form is "3840 pixel-widths wide x 2160 pixel-heights tall = a bazillion pixel-areas" (and one "pixel area" is definitionally equal to "pixel-width * pixel-height" for rectangular pixels).
N.B. it is frequently assumed that pixel width and pixel height are equal (as in the CSS discussion in the other answer), and even without that assumption the above assumes that pixels are rectangular -- and these assumptions are often but not always true!