the [sides/facets/dimensions] of a problem.
Is there any difference between sides/facets/dimensions? Which one is more common and idiomatic?
the [sides/facets/dimensions] of a problem.
Is there any difference between sides/facets/dimensions? Which one is more common and idiomatic?
When speaking about literal geometric shapes, we normally use the word 'sides'. With 3-dimensional shapes, we use the word 'faces', although 'sides' may also be used.
Still speaking literally, 'facet' is used to refer to the sides or faces of a cut gem, such as a diamond. It isn't normally interchangeable with 'sides' or 'faces' in this physical context. For example, a fixed object such as a building may have a 'left side' or a 'north face'. There are lots of reasons you wouldn't refer to any of the surfaces of a multi-faceted diamond in such a fixed way.
Your context is figurative though - and in this context, either could be used. Just remember that 'facets', like its literal use, suggests multiple aspects. Saying that a problem or issue is 'multi-faceted' suggests that it is complex. By contrast, it is common to say that there are "two sides" to something.
'Dimensions' does not have the same literal meaning as any of the above, but in a figurative sense it can mean 'properties', which in turn can be used to refer to different aspects of something. You could use it in your example but it may not be as common. It better suggests that there are different depths to something rather than different ways of looking at it.