I even guess at words by taking what I think is an English word with a latin root, and converting it to Spanish, which sometimes works really well and at other times it doesn’t work quite so well at all!
"Periphery" does not have Latin etymology. "Peri" is overtly Greek, and so is "ph".
The Latinate version would be "circumference".
Yes, the English word (and the Portuguese cognate, periferia) does have a Latin etymology, via Late and Medieval Latin peripheria.
Now, the Latin word peripheria itself has a Greek etymology, from Greek periphereia. But both Portuguese and English get their word via Latin (and, in the English case, possibly also French, which also gets it from Latin, though sources differ on whether English got it from French or directly from Medieval Latin, and its plausibly not uniquely one or the other.)
so it's not that that word isn't fancy in portugese too. It's that that word might come easier to a portugese (or other romance language) speaker, since it's a cognate, so they use it, and end up sounding fancy in English.
An English speaker could do the same thing in portugese with the same words in reverse!