The smallest circle that two parts of a semi-circle can fit into?

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SUMMARY

The discussion centers on determining the smallest circle that can contain two parts of a semi-circle, specifically when cutting a circular tortilla. Three optimal solutions are proposed: the first requires a straight-line cut from the center to the edge, the second allows for any division of the semi-circle, and the third permits non-linear cuts. The suggested method for achieving the smallest plate size involves drawing a circle with a radius of $\frac{r}{\sqrt{2}}$ and consuming the annulus first, leaving minimal leftovers.

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  • Explore geometric optimization problems involving circles and semi-circles.
  • Research the properties of annuli and their applications in geometry.
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Mathematicians, geometry enthusiasts, culinary artists interested in food presentation, and anyone involved in optimization problems related to shapes and space utilization.

TrevorE
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So, true story:

I made a large circular tortilla.

Ate half of it. Then decided to put the rest into the fridge on a smaller plate.I raised the knife to cut the remaining semi-circle in two, and then went : "Hmmmmmmmm...".

Anyway, it's in the fridge now with an approximate solution, but I'm wondering if anyone knows the mathematical one?

I'm wondering if there are 3 different optimal solutions:

1) In which the straight-line cut necessarily is from the center of the original circle to the edge. I.e. a radius.
2) In which the straight-line cut may divide the semi-circle in any possible way.
3) In which the cut is not necessarily a straight line.

Thanks in advance if anyone can offer a demonstration.View attachment 9517
 

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Next time, don't cut it that way. Instead, draw a circle on your pizza with radius $\frac r {\sqrt 2}$ and eat the annulus for your first half. You will be left with the smallest possible plate size.
 

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