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This question already has an answer here:

Why can't I loop cut a cone? I started a new file, created a cone and did nothing but try to loop cut it, and it didn't work.

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marked as duplicate by David Sep 29 '15 at 23:03

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In layman terms loop cut will only work on geometry with 'clear edge flow' meaning in most if not all cases you should ideally be able to look at it and tell where the cut will go or it should divide your mesh equally into more sections that can be cut again. So it won't work for triangles (3 vertices) or ngons (mesh with more than 4 vertices), What is non-manifold geometry?. There is a simple work around however. Create a cone by adding a cylinder and scaling the top as small as possible without actually letting the vertices overlap, when zoomed out, this is unnoticeable.

enter image description here

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    $\begingroup$ Once you're finished, it is helpful to remove doubles to keep your vertex count and processing load down. $\endgroup$ – Michael Eric Oberlin Sep 2 '17 at 0:34

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