In the Immensikoff thread, iakon mentioned that according to the Online Etymological Dictionary pediment is “apparently a dialectal garbling of pyramid.” If you had asked me where it came from, I would have said “I suppose a Latin pedimentum,” but as it turns out there is no such word in Latin. Here’s the OED’s etymology (updated September 2005):
Origin uncertain; perhaps alteration of pyramid n. (compare pyramid n. 3a), based on a colloquial or regional pronunciation of the word (compare forms s.v.), with assimilation of the final syllable to -ment suffix.
An alternative derivation as an aphetic form of operiment n. ["A covering," from Classical Latin operīmentum covering, cover < operīre to cover] has also been suggested. In quot. 1592 at sense 1aα. ["The Coronices..were corrospondent and agreeing with the faling out of the whol worke, the Stilliced or Perimeter [Margin. A periment in corrupt English]“] the word is considered an alteration of perimeter n.; although supported by an isolated Middle English variant perimentre of perimeter n., the development is unlikely for semantic and phonological reasons.
The β forms [i.e., with ped- rather than per-] show assimilation of the first element to classical Latin ped- , pedi- [...]; compare classical Latin pedāmentum stake or prop for vines and later pedament n., also Italian †pedamento foundation, groundwork, base, footing (1499, 1611 in Florio). The form pedament perhaps shows influence of Italian pedamento; the form piedment probably shows influence of French pied (see pied-à-terre n.).
The association of the first element with Latin ped- also influenced the semantic development (see sense 2). Compare pedestal n.
The first citation after the 1592 alteration of perimeter is from 1601–2 (“and for a periment in the middest of the same wanscott xxs”), and the first with the ped- spelling is from 1664 (J. Evelyn Acct. Archit. in tr. R. Fréart Parallel Antient Archit. 140: “Those Roofs which exalted themselves above the Cornices had usually in face a Triangular plaine or Gabel (that when our Workmen make not so acute and pointed they call a Pedament) which the Antients nam’d Tympanum”).
Such a simple-looking word to conceal such mysteries!