If $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=1$  and $z_1+z_2+z_3=0,$  then area of the triangle whose vertices are $z_1,z_2,z_3$  is

$\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=1$

Hence, the cirucmcemntre of triangle is origin, Also, centroid $(z_1+z_2+z_3)/3=0,$  which coincides with the cirucmcemntre.

So the triangle is equilateral. Since radius is 1, length of side is $a=\sqrt{3}.$  

Therefore, the area of the triangle is $(\sqrt{3}/4)a^2=(3\sqrt{3}/4).$