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I'm trying to convert a polygon shapefile of the UK into a hex grid. Using MMQGIS I've produced the grid, then I tried the intersect tool which gave me something close to what I want. However, the irregular edge of the UK borders has cut some of the hexagons. Is there a way of keeping all of the hexagons whole and producing an 'approximation' of the original polygon.

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You actually want a selection of the grid rather than a geometric modification of it so use Select by Location (one menu lower down under Vector Selection Tools) and chose intersects and contains to pick up all the hexagons that are within and overlapped by the UK polygons. Then you'll need to save the selection to a new file of course.

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