How long have you had the phone? If you had it for a while, chances are you need a new battery. It's possible to replace the battery yourself (cheapest way, but can be hard to do) but I would recommend going to a shop and have them do it (possibly a little most costly).
Yeah, this can happen especially if the iPhone is over 1.5-2 years old. I'd say its not worth replacing the battery, either get a new phone (ouch) or purchase a large battery case that can essentially "take over" for your battery. Maybe something like this: