I need to understand how to find the area of circle inscribed in a hexagon with the least amount of waste.

If you have a regular hexagon, and you know the length of a side then you can find the area of both.

the area of the hexagon (using the length of one of its sides) is:

(3s²√3)/2 sq.units, where s is the length of the side.

The largest circle that can be inscribed will have a radius of (s√3)/2 so its area will be:
(3πs²)/4 sq. units.

You can also now find the amount of waste by subtracting the area of the circle from the area of the hexagon.

hope that helps.
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