I am refurbishing a small cabinet and the door has this type of hinge. However, the peg was broken off of one of them and I can't seem to find a replacement. What is this type of hinge called?
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This site is not normally a shopping helper site but I took a look to see if it that style of peg hinge popped up as a common type. (I have seen these in the past believe it or not). Most of the typical suppliers to cabinet makers (Rockler, Woodcraft, McFeeleys) do not appear to carry hinges like that.
The closest I could find (and it may work for you) is a metal peg style shelf support.
Metal shelf support pegs are available in many of the big box stores and online at places including Amazon.
Edit: Rockler does offer a similar type of hinge that may be adaptable to your application. These have the advantage in that there is some adjustability of the door alignment and the doors are easily removable.