Learn Russian: Кот спит около угла этой стены. - The cat is sleeping near the corner of this wall.

The preposition около (near) always requires the Genitive case. The dictionary form is угол (masculine). When it changes to Genitive, the second о drops out (this is called a fleeting vowel) and we add the standard masculine Genitive ending -а, making it угла.

Why is 'this wall' translated as этой стены?

In Russian, you express 'of something' by using the Genitive case instead of a preposition. Since стена (wall) is feminine, it becomes стены. The word for 'this' must match the gender and case of its noun, so the feminine эта changes to its Genitive form, этой.

So both угла and этой стены are in the Genitive case?

Exactly! They are Genitive for different reasons. Угла is Genitive because it follows the preposition около. Then этой стены is Genitive to mean 'of this wall'. Chaining Genitive words like this is very common in Russian.

Is этой only used for the Genitive case?

No, it is actually the form used for almost all cases of feminine nouns! For the feminine word эта, the Genitive, Dative, Instrumental, and Prepositional forms are all exactly the same: этой. In this sentence, we know it is functioning as the Genitive because it translates to 'of this'.