Its a fair distance between the Circular Quay central area to the Opera House, it's quite nice to take the time to walk to the Botanics Garden and have a walk around. We went there in between time before our show starts in the evening, quite a nice view and a few attractions to notice in the Botanics Garden.
One thing to note is the gate for the garden closes quite early though, they shuts off at around 5.30pm, so do go there a little earlier, I think once you are in, you are able to stay in there a little after the gate closing time, the exit gate should be open all the time, just that the gate to enter has a time limit (we didn't try though, so.. no guarantee =P, well we didn't see any staff of management the whole time there)
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