What are some relations between the golden number or ratio $\phi$, and $\pi$?
For example, by considering this answer https://math.stackexchange.com/a/744196/ ; by Steve Lewis.
Now taking the point at $\pi$ and not $1/2$, one can notice that at the half of the hypotenuse of the triangle, it's aproximately $\phi=\dfrac{1+\sqrt5}{2}\approx1.618$
As @BrianTung correctly points out, your diagram shows nothing but $\varphi \approx \pi/2$, to questionable accuracy. If you want to prove something by geometric means, measuring out numerical approximations of irrational quantities certainly is not the way to go about it. This is not to say that you cannot prove facts involving irrational quantities geometrically, but such proofs make use of other properties of those quantities. Now to the point. The golden ratio is an algebraic number, namely the positive solution of the quadratic equation
$$ x^2 -x - 1 = 0 $$
while $\pi$ is transcendental, meaning there is no polynomial with coefficients in $\Bbb Z$ such that $\pi$ is a root of this polynomial. This means that it is impossible to construct one from the other using elementary arithmetic and radicals, or by a compass and straightedge. However, the following identity holds $$ e^{i\pi/5} + e^{-i\pi/5} = \varphi $$
following from two facts: The complex tenth roots of unity form a regular decagon, in which two regular pentagons can be inscribed, and the that the ratio of the side length and the diagonal of such a regular pentagon is precisely $\varphi$.
The OP wishes to know if there are relations between pi and phi. First, recall the well-known,
$$\pi \approx \frac{\ln\left(640320^3\,+\,744\right)}{\sqrt{163}}$$
which differs by a mere $10^{-31}$. However, we can also use the golden ratio in a similar approximation,
$$\begin{align} \pi &\approx \frac{\ln\left((2^6\phi^6-24)^2-552\right)}{2\sqrt{5}}\\[5pt] \pi &\approx \frac{\ln\left((2^6\phi^{12}\color{red}+24)^2-552\right)}{2\sqrt{10}}\\[5pt] \pi &\approx \frac{\ln\left((2^{12}\phi^8-24)^2-552\right)}{2\sqrt{15}}\\[5pt] \pi &\approx \frac{\ln\left((2^6\phi^{24}-24)^2-552\right)}{2\sqrt{25}} \end{align}$$
which differs by $10^{-7},10^{-11},10^{-14},10^{-18}$, respectively. While the first example used the j-function, the latter ones used the Dedekind eta function and which explains their consistent form.
A consequence is that while $1/\pi$ can be expressed a sum of negative powers of $640320^3$, then $1/\pi$ is also the sum of negative powers of the golden ratio in at least four ways, the fourth as,
$$\frac1{\pi}=\frac{10}{5^{1/4}}\sum_{n=0}^\infty\frac{(2n)!^3}{n!^6}\,\frac{6(18\phi-29)n+(47\phi-76)}{(2^6\phi^{24})^n}$$
Therefore, there are non-trivial relations between pi and phi.
P.S. Note the appearance of the Lucas numbers $L=2, 1, 3, 4, 7, 11, \color{red}{18, 29, 47, 76},\dots$ (cousins of the Fibonacci numbers) in the pi formula above.
