The curve is symmetrical with respect to the origin, and occurs only with values of θ from -45° to 45° (-¼ π to ¼ π).

The area in polar coordinates is:

$A = {\displaystyle \frac{1}{2}{\int_{\theta_1}}^{\theta_2}} r^2 \, d\theta$

$A = 4 \left[ {\displaystyle \frac{1}{2}{\int_0}^{\pi/4}} a^2 \cos 2\theta \, d\theta \right]$

$A = 2a^2 \left[ \dfrac{1}{2} \sin 2\theta \right]_0^{\pi/4}$

$A = a^2 [ \, \sin \frac{1}{2}\pi - \sin 0 \, ]$

$A = a^2 \, \text{ unit}^2$ *answer*