Learn Russian: Я взял книгу квадратного формата на неделю. - I took a book of square format for a week.

This is the genitive of qualitative description. It acts like an adjective to describe the book's inherent physical trait. Just like in English where you can say a book of a square format, Russian uses the genitive case (квадратного формата) to attach this descriptive phrase to книгу.

Could I just say квадратную книгу instead?

Yes, you absolutely can. Я взял квадратную книгу means "I took a square book" and is very common. Using the genitive descriptive phrase квадратного формата sounds slightly more formal or focuses specifically on the book's layout and dimensions.

Why do we use на неделю instead of just saying неделю to mean "for a week"?

When you want to express how long the result of an action is intended to last, you use на + the accusative case. If you just used the accusative without a preposition (неделю), it would mean the action itself took a week to complete (i.e., you spent seven days in the physical act of picking the book up).

Is взять на неделю the standard way to say "to borrow for a week"?

Yes! Russian doesn't have a single, universal verb for "borrow" that fits every situation. For physical objects like a book, using взять (to take) plus на and the intended time period is the most natural way to express borrowing.