I'm not sure what you mean by "formal definition," but here are the dictionary definitions:
1. a measurable extent of some kind, such as length, breadth, depth, or height.
"the final dimensions of the pond were 14 ft. x 8 ft"
synonyms: proportions, measurements, extent, size; More
2. an aspect or feature of a situation, problem, or thing.
"sun-dried tomatoes add a new dimension to this sauce"
synonyms: aspect, feature, element, facet, side
"water can add a new dimension to your garden"
1.form or shape (something) to particular measurements.
"the seats and backrests are dimensioned to withstand high loads from items such as unsecured luggage"