In microns
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An instrument called micrometer is used to measure paper or card thickness.
In typesetting, the thickness of a line is called its weight and is measured in points.
mils
To measure the thickness of a thin piece of wire we have got the instrument called the wire gauge. The thickness of a thin piece of wire is measured in mm.
"Thickness" is a length or distance. The SI unit for length or distance is the meter. In order to produce a convenient number when you're talking about the thickness of a piece of paper, you'll probably want to use one of the meter's sub-multiples, such as the centimeter or millimeter. For tissue or 'onionskin', you might even go to the micrometer.