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.
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Surface area of three dimensional figures are measured in square units whereas their volumes are measured in cubic units
Square units are used to measure the area of something.
Area = ( length of a side)2
Surface Area is technically the same as area if we are dealing with 2-dimensional surfaces. But, it is mostly used to find an area of a 3-dimensional object.
You could use h*BA cubic units, provided that the units used for BA are squares of units used for h. If BA is in square centimetres and h is in metres, the formula will give a answer which is not easy to understand.