Learn Russian: Это крошечная тетрадь квадратного формата. - This is a tiny notebook of a square format.

This is the genitive of qualitative description. It acts like an adjective to describe an inherent characteristic of the noun before it (тетрадь). In English, we often say 'a notebook of a square format', which matches this Russian structure perfectly.

Could I just say квадратная тетрадь instead of using the genitive квадратного формата?

Yes, квадратная тетрадь (a square notebook) is perfectly natural and grammatically correct. Using тетрадь квадратного формата is just a slightly more formal or precise way to describe it, focusing on its physical dimensions or paper size.

Why does the sentence use формата instead of формы (shape)?

While квадратной формы (of a square shape) is also completely valid, формата is frequently used for paper goods, books, and notebooks. It implies standard printing dimensions or layouts, much like how we refer to 'A4 format' in English.

Тетрадь ends in a soft sign. How do we quickly remember that it is feminine and requires крошечная?

Most Russian nouns ending in -дь (like тетрадь, площадь, очередь) are feminine third-declension nouns. Because it is feminine and in the nominative case, the modifying adjective must be the feminine nominative крошечная.