thanks for trying, but the correct answer I found out on my own. I guess ill post an explanation here
We take a cross-section along the great circle. Let O be the center of the sphere, and let M be the midpoint of line AB
Since OA = OB = 1, angle OMA = 90, Am is sqrt3/2 so triangle AOM is a 30 60 90 triangle
So AOM is 60 degrees, BOM is also 60 degrees, therefore AOB is 120 degrees, therefore arc AB has length
120/360 *2pi=2/3pi
+180 