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I have created a shapefile in ArcCatalog by right-clicking the folder I want it in, then selecting new, then shapefile, then changing the type to "polygon", and using the British National Grids under the projected co-ordinate system.

I then tried to add the shapefile in ArcMap by clicking the Add Data button and selecting the shapefile I created.

My problem is that when I open the shapefile in ArcMap, the map is still blank and I cannot see the projected shape of the British National Grids.

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You need to generate your own grid after setting the projection. You can download one from the GISSE answer here:… – Mapperz Jan 3 '13 at 14:17
@Mapperz - The answers on there are a prime example of where answers stagnate over time. I.e. The link in the top-rated answer is now a 404. – GIS-Jonathan Jan 4 '13 at 12:00 - go there and download Strategi for a 50km grid. I don't think any of the other products include a grid. – GIS-Jonathan Jan 4 '13 at 12:06

Specifying "British National Grids" in the projection sets the projection of your data (regardless of what data your shapefile will contain). It DOES NOT populate the shapefile with vectors/polygons of the national grids.

In order to see the British National grids as polygons, you need to either find a source that already has the data, or digitize the grids yourself. The answers to Ordnance Survey grids - download or generate? include possible sources for the data.

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