Given a convex polygon, and a smaller hexagon. How to estimate the maximal number of those hexagons that is necessary to cover the given polygon?

There are three cases with different bounding conditions.

  1. In the first case all hexagons should be contained inside the polygon. enter image description here

  2. In the second case hexagons can intersect the borders of the polygon, but the centers of hexagons should be contained inside it.

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  1. The third case continues the second one. Hexagons also can intersect the borders of the polygon, but the circles with the given radius, located in the center of every hexagon, should be placed inside the polygon. (All circles are equal, and they are smaller or equal to the inscribed circles.)

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The only solution I found is to divide the area of the polygon into the area of the hexagon, but it is approximate.

Update 1

All hexagons are always stick together, or, in other words, they form a hexagonal grid. That grid could be rotated and translated in any way.

Update 2

May be it could be easily to think about convex polygon which covers a regular hexagonal grid.


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