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Either could be but the conversion from length to area and volume, in terms of units, is simpler.

Area = Length2 and Volume = Length3.

If Area were the base, then

Length = Area0.5 or sqrt(Area) and Volume = Area3/2 - not a convenient relationship.

If Volume were the base, then

Length = Volume1/3 or cubert(Volume) and Area = Volume2/3 - again not a convenient relationship.

Length is required far more often than area or volume in other dirived measures such as speed, acceleration, force, power and so on. So having to calculate roots is making life complicated uncessarily!

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Q: Why area and volume are volume are not considered as base units although they depend upon length?
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