The normal at $P$ cuts the axis of the parabola $y^2=4ax\text{ in }G\text{ and }S$  is the focus of the parabola. If $\mathrm{\Delta }SPG$ is equilateral then each side is of length

Let  $P\left(a{t}^2,2at\right)$

Equation of the normal at $P\left(t\right)\text{ is }y+tx=2at+a{t}^3$

Equation to y-axis is  $x=0$. Solving  $G\left(2a+a{t}^2,0\right)$

Focus  $S(a,0)$

$\begin{array}{l}\mathrm{\Delta }SPG\text{ is equilateral }\Rightarrow PG=GS\Rightarrow \sqrt{4a^2+4a^2{t}^2}=\sqrt{a^2}{\left(1+t^2\right)}^2\\ \Rightarrow 4a^2\left(1+t^2\right)=a^2{\left(1+t^2\right)}^2\Rightarrow 4=1+t^2\Rightarrow t^2=3\end{array}$

Length of the side  $=SG=a\left(1+t^2\right)=a(1+3)=4a$.