Answer
$T o t a l\;l e n g t h=4*\frac{3a}{2}=6a$
Work Step by Step
$$
\begin{gathered}
\frac{d x}{d \theta}=-3 a \cos ^2(\theta) \sin (\theta) \\
\frac{d y}{d \theta}=3 a \sin ^2(\theta) \cos (\theta)
\end{gathered}
$$
$$
\begin{gathered}
\left(\frac{d x}{d \theta}\right)^2+\left(\frac{d y}{d \theta}\right)^2=9 a^2\left[\cos ^4(\theta) \sin ^2(\theta)+\sin ^4(\theta) \cos ^2(\theta)\right] \\
=9 a^2 \sin ^2(\theta) \cos ^2(\theta)\left[\cos ^2(\theta)+\sin ^2(\theta)\right] \\
=9 a^2 \sin ^2(\theta) \cos ^2(\theta)
\end{gathered}
$$
$$
\begin{gathered}
\sqrt{\left(\frac{d x}{d \theta}\right)^2+\left(\frac{d y}{d \theta}\right)^2}=\sqrt{9 a^2 \sin ^2(\theta) \cos ^2(\theta)}=3 a \sin (\theta) \cos (\theta)=\frac{3}{2} \mathrm{a} \sin (2 \theta) \\
s=\int_0^{\pi / 2} \frac{3 a}{2} \sin (2 \theta) d \theta=-\frac{3}{4}[\cos (\pi)-\cos (0)]=\frac{3 a}{2}
\end{gathered}
$$
Total length $=4 * \frac{3 a}{2}=6 a$
