As can be seen from the figure, OD=OE=OF, BD=0.5, angle ODB = 90 and angle OBD = 30°, so OB = 2OD.

In the Rt triangle of OBD, OD square +BD square =OB square.

Square of 3d =0.25

OD square =112

Outer diameter =√3/6

So the total length is √3/2.