In the UK simple timber-members supporting floors/roofs spaced at 16" c/c [400mm] the rule of thumb is for a width of ~2" [50mm] the depth is span_in_feet_as_inches/2 +1; so a joist spanning 14' [4.270m] needs to be 14/2 + 1 = 8" [200mm] - if you use the reg-tables or actually calculate it that is what you'll get !
For most steel-beams we'd take 1/20 span as the initial design depth, so generally a 12" U-beam spans 20' [300mm for 6m].
Obviously these things need calculating properly, but this is useful as an early design indicator - indeed, the standard way of designing a steel-beam in the U K is to 'guess its size', then check that with its live+dead-loads for adequacy in stress/shear/deflection, then if it's under-stressed you 'go down', or over-stressed you 'go up': there are several 'weights/linear m' of most standard steel beams which are nominally the same overall sizes but have differing wall/web/flange thicknesses - since you pay 'by the ton' it can be cheaper to use a deeper beam with thinner cross-sections, as that may well be cheaper than a heavier beam that is smaller - of course is you are stuck for space you may well have to use the more expensive one !
It is difficult to estimate the depth of 'engineered' timber or steel members; however, the manufacturers will usually give tabular indications of spans/loads >> depths etc... A lattice beam may well be deeper than a solid beam for the same span, but of course the lattice version will contain considerably less material, so balanced against its fabrication/connector and additional external envelop costs it can well be cheaper than the solid equivalent - and of course the heavier solid one can have knock on affects on other members/foundations etc, and it is also less suitable for passage pipes and ducts within its depth...