The term elegant hairstyles means different things to different groups of people. For one group of people, for anything to qualify to be considered as part of the elegant hairstyles, it has to be a 'formal hairstyle' in the truest sense of the word; so that for those who subscribe to this way of thinking, elegant hairstyles automatically translate to 'formal hairstyles' and the two terms can thus be used interchangeably. Another school of thought has a less stringent criteria, so that any hairstyle that is reasonably neat, whether among what are considered formal hairstyles or not, would still be termed as elegant.
For the most part, elegant hairstyles are characterized by short to medium sized hair - though with a lot of care, longer hair can still be worked into one of these elegant hairstyles. There are a number of specific hairstyles that are almost universally considered to be in the category of so-called elegant hairstyle.
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Buns are one example of hairstyles that are almost universally considered to be among elegant hairstyles. The idea in these buns, also called chignons, is to gather the hair towards the back of one's head, and then putting together the bun with a pin. Of course, to make the bun elegant, that hair first has to be 'softened' through a procedure like blow dry, and then have all sides of the 'bun' combed lightly, to keep it all in. properly done, this can be a very elegant looking hairstyle, and it is indeed one of the cheapest elegant hairstyle that one can achieve without having to spend a fortune on the process.
Talking of elegant hairstyle, it is worth mentioning that there are occasions when wearing such elegant hairstyles becomes a necessity - that is, something you actually have to do to 'look proper.' Bridal parties, for instance, typically take time to acquire the elegant hairstyles so as not to let down the bride on her 'big day' - as do people going to a prom or some other sort of event where elegance is demanded of the participants.