kvadrato

The false seam first. Kvadrato ends in -at-, which is a real Esperanto suffix: the present passive participle, as in amata (being loved) and vidata (being seen). Read the word that way and you would need a verb kvadri, which does not exist, and you would get "a thing being four-ed", which is not a shape. Kvadrat- is one indivisible root and the participle reading is an accident of spelling.

The other half of the suspicion is right, though. Kvar really is behind it. Latin quattuor (four) gave quadrare, to make four-cornered, whose participle quadratus is our root; and the same quattuor gave Esperanto kvar directly. So the two words are cousins, arriving in the language by different routes and centuries apart. That is the pleasant version of the etymology trap: the instinct is correct about the history and wrong about the morphology, which is the same shape as inspiri, konsterni and reklamo.

Not the one in the town centre. English uses square for the shape and for the open space in front of the town hall; Esperanto does not. The public square is placo, from Latin platea, and it is a different word with a different history. La ĉefa placo de la urbo is the main square; kvadrato there would mean somebody had drawn a rectangle on the map.

In use. Kvadrata is the adjective, so kvadrata metro is a square metre and kvadrata radiko a square root. Kvadratigi is to square something, either geometrically or arithmetically, and la kvadrato de kvin estas dudek kvin. Kvadrateto is the small square on graph paper or on a chessboard, and kvadratita papero is squared paper. The neighbouring shapes are rektangulo, triangulo, rondo and cirklo.