I did not have a knife with me, so I solely used my unit circle cookie cutter to separate every sq. like this:

I then rearranged the elements into this form:

Since the angle coated by this form is strictly 120 levels (see the ultimate spoiler block to verify), three of those make a pleasant circle, with some white shining by the gaps:

Since the match of the inexperienced piece into the indentation within the purple one is a bit cosy (it is a precise match, really), let’s verify that there isn’t any overlap:

The hole within the purple piece and the tallest a part of the inexperienced piece each have the peak of “1 minus the peak of an equilateral unit triangle”, so the match is actual, and we’re accomplished.