I looked up in the dictionary and found they both mean the boundary of a closed curve, or the extended meaning “not the center”. Are they synonyms?
They have a close meaning and can be synonymous in some cases. The New Oxford American Dictionary lists one of the meanings of perimeter, “the outermost parts or boundary of an area or object,” which is also the main definition of periphery.
However, the most usual meaning of perimeter is geometrical: it means the line delimiting a figure or an object.
I'm going to call these words unrelated (this is not going to be the "credited response" on a multiple choice test, but please hear me out.) Here's an simple example to illustrate my thinking. Suburbs lie outside of city centers. There may be many suburbs outside of a city center at varying distances. In every case, though, suburbs are "peripheral" to the city, that is, they are in an area outside of, but conceptually related to, the city center. "Conceptually related to" is a fluid idea that may encompass geographic distance, commute times, home values, etc. On the other hand, a state border is a defined, measurable boundary, or perimeter. The conceptual "periphery" of a city center may lie within or outside of the city's state border, or perimeter. The NY Metro area, for example, includes peripheral suburbs in NJ and CT. The "perimeter" of New York State is not related to the configuration of the Metro, nor is the state border a mere concept used to describe any place outside of but conceptually related to New York City. "Periphery" is the word that gets that job done.
Is this clear yet? If not, read on.
Periphery? Always used to refer to an object's location anywhere outside of a point of reference - either physical or metaphorical. For example, "This discussion is on the periphery of the real topic at hand." Or, "I was looking at the street light, but I could see a car coming from the left in my peripheral vision."
Perimeter? A DEFINED BOUNDARY drawn around an area.
This is a tricky one, and certainly interesting. But I'm going with NOT (usually, commonly, or necessarily) RELATED.