Approximations of $\pi$ using radicals There are many approximations of $\pi$ using trigonometric and rational numbers. But I created this one: $$\pi \approx \sqrt[11]{294204}$$ Which is correct to almost $8$ decimal places. Are there any other approximations of $\pi$ using radicals? I know of $\sqrt{10}$, $\sqrt[3]{31}$, $\sqrt[4]{97}$, and so on. But are there more beautiful ways (like using $\varphi$ since it is composed of radicals)?
 A: There are lots of radical approximations to $\pi$. A classical example is given by Viete's formula, in which a series of increasingly complicated nested radicals may be used to render $2/\pi$:
$2/\pi=\Pi_{n=1}^\infty(a_n/2)$
$a_1=\sqrt2, a_n=\sqrt{2+a_{n-1}}.$
Many readers are aware that this series is derived by considering the perimeters of inscribed regular polygons having $2^k$ sides, these approaching the circumference of a circle as $k\to\infty$.
Just for fun, let's explore a method based on the Taylor series for the cosine function. Pick a value of $n$ for which $\cos(\pi/n)$ has a convenient radical expression and plug in the Taylor series approximation
$\cos(\pi/n)\approx 1-[(\pi/n)^2/2]+[(\pi/n)^4/24].$
Then substitute the radical expression for $\cos(\pi/n)$ on the left and solve the biquadratic polynomial by standard methods for "$\pi/n$".
With $n=5,\cos(\pi/5)=(\sqrt5+1)/4$ this gives
$\pi\approx5\sqrt{6-\sqrt3×(\sqrt5+1)}=\color{blue}{3.142}3...$
(correct to the significant digits indicated in blue.)
A: COMMENT: By way of simple pertinent information.
There are many remarkable approximations of $\pi$. For example
$$\pi\approx\dfrac{22}{17}+\dfrac{37}{47}+\dfrac{88}{43}\\\pi\approx\sqrt[4]{\frac{2143}{22}}\\\pi\approx\sqrt[9]{\frac{34041350274878}{1141978491}}\\\pi\approx\frac{\ln(640320^3+744)}{\sqrt{163}} $$
The first and second each give $9$ exact decimal places, the third gives $15$ and the fourth gives $30$.
Buffon's experiment.- On any flat surface covered by parallel straight bands of equal width, a rod of length equal to the width of the bands is pulled a given number $T$ of times. In each shot the rod either stays within a band or cuts one of the parallel delimiting lines. Let $C$ be the number of cuts obtained; the quotient $\dfrac{2T}{c}$ it will be in principle the closer to $\pi$ the higher the number $T$.
In 1901, the Italian M. Lazzarini using a rod with a length equal to $\dfrac56$ of the width of the bands (in which case the theoretical probability of cutting is $\dfrac{5\pi}{3}$, he got in 3408 shots the rational approximation $\dfrac{355}{113}\approx3.141592$ .
A: The well-known $\pi \approx \frac{\ln\left(5280^3\,+\,744\right)}{\sqrt{67}}$ and $\pi \approx \frac{\ln\left(640320^3\,+\,744\right)}{\sqrt{163}}$ involves the j-function and integers. But we can also use the Dedekind eta function and radicals,
$$\begin{align}
\pi &\approx \frac{\ln\left(2^6\phi^6-24\right)}{\sqrt{5}}\\
\pi &\approx \frac{\ln\left(2^6\phi^{12}+24\right)}{\sqrt{10}}\\
\pi &\approx \frac{\ln\left(2^{12}\phi^8-24\right)}{\sqrt{15}}\\
\pi &\approx \frac{\ln\left(2^6\phi^{24}-24\right)}{\sqrt{25}}
\end{align}$$
with golden ratio $\phi =\frac{1+\sqrt5}2$. The last two are correct to $9$ and $12$ decimal places, respectively.
P.S. The consistent appearance of $24$, the powers that are factors of $24$, and the radicands that are multiples of $5$ are clues the above are not coincidences and in fact have a mathematical reason behind them. Lovely, aren't they?
