In [this link][1], Corollary 3.2.2, page 59 the author claims that: The Euler characteristic of the toric variety $X_K$ associated to a convex polytope $K$ is the number of vertices of $K$.

I want to see how it works.  Could someone please illustrate this for me by using this method to compute the Euler characteristic of $\mathbb{P}^{2}$ and $\mathbb{P}^{1}\times \mathbb{P}^{1}$?  Thanks so much.

[1]: http://www.math.leidenuniv.nl/scripties/Trevisan.pdf