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It depends. If the piece of paper is 1 micron wide and 10 kilometres long, it will have an area of 0.01 sq metres but I would not be able to write anything on paper that narrow. Or, worse, it could be 1 nanometre wide and 10,000 km long and still have an area of 0.01 sq metres!
Square metres. Square centimetres for small objects like a sheet of paper, square kilometres for a town or lake.
A sheet of A4 paper has an area of 1/16 square metres. Its dimensions are approximately 297 mm x 210 mm.
A4 is a standard paper size (1/16 square metres). The standard for printer paper is 80 gsm (grams per square metre). An 80 gsm sheet of A4 paper weighs 5 grams.
Square kilometres for lakes or towns, square metres for rooms or gardens, square centimetres for pictures or sheets of paper are some common units and the magnitudes that they would be used to measure.