<s>[INST] What are better approximations to $\pi$ as algebraic though irrational number? I know that $\pi \approx \sqrt{10}$, but that only gives one decimal place correct. I also found an algebraic number approximation that gives ten places but it's so cumbersome it's just much easier to just memorize those ten places.
What's a good approximation to $\pi$ as an irrational algebraic number (or algebraic integer if possible) that is easier to memorize than the number of places it gives correct?
EDIT: Algebraic number preferably of low degree, such as $2$ or $3$ (quadratic or cubic).
 [/INST] Dalzell's integral is related to the rational approximation $\pi\approx \frac{22}{7}$.
$$\pi=\frac{22}{7}-\int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx\approx\frac{22}{7}$$
Similar small integrals are related to simple irrational approximations using $\sqrt{2}$ and $\sqrt{3}$.
$$\pi=\frac{20\sqrt{2}}{9}-\frac{2\sqrt{2}}{3} \int_0^1 \frac{x^4(1-x)^4}{1+x^2+x^4+x^6}dx\approx \frac{20\sqrt{2}}{9}$$
$$\pi=\frac{9\sqrt{3}}{5}+\frac{6\sqrt{3}}{5}\int_0^1\frac{x^3(1-x)^2}{1+x^2+x^4}dx\approx \frac{9\sqrt{3}}{5}
$$
Fractions $\frac{20}{9}$ and $\frac{9}{5}$ are convergents of $\frac{\pi}{\sqrt{2}}$ and $\frac{\pi}{\sqrt{3}}$ respectively.
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