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.