Can someone tell me where is the error in his reasoning? (Sorry for the pessimistic viewpoint)
The triangle he refers to does not have one half of the area of the circle, as he claims. This is where the error comes from.
In the future, I might recommend to not bother reading sites that claim obviously false things, like $\pi=3.15...$.
I'm not sure he is trying to say that his estimate of $\pi$ is better, just that this construction provides a reasonably good estimate. And it does, but we know this only because we know a good approximation of $\pi$ beforehand (that is, we can compare the calculated value to 3.1415926...)
His error estimate makes me wonder though, since it's way off. If he is trying to claim that his estimate of $\pi$ is better than the usual 3.1415926..., then he would need to better explain his error estimate, which comes out of nowhere.