Your idea matters-2

$\begin{aligned} 4\left(\left(\frac 12\right)! \right)^2 & = 4\left(\Gamma\left(\frac 32\right) \right)^2 & \small \blue{n! = \Gamma (n+1) \text{, where }\Gamma (\cdot) \text{ denotes the gamma function.}} \\ & = 4\left(\frac 12 \Gamma\left(\frac 12\right) \right)^2 & \small \blue{\text{Note that } \Gamma (1+s) = s\Gamma(s)} \\ & = 4\left(\frac {\sqrt \pi}2 \right)^2 & \small \blue{\text{and } \Gamma \left( \frac 12 \right) = \sqrt \pi} \\ & = \pi \approx \boxed{3.14} \end{aligned}$

Reference: Gamma function

$4\times \left(\left(\frac 12\right)!\right)^2=4\times \left(\frac{\sqrt{\pi}}{2}\right)^2=\boxed{\pi}.$