First off, cones always have a front width the same as the cone length, and the front of a cone is always flat, not curved.

Xanathar's Guide outlines two ways you can apply AOEs on the battlemap.

The template method, where you basically overlay a template on the grid, and all squares touched by the template are affected. UNLESS the template is a sphere, in which case squares covered 50% or more by the template are affected.

And the Token Method, where you basically get what Matt Colville meant when he called 5e "non-euclidian": 5" squares are 5" in all directions. In this case you put out d6's or other tokens, and keep the length in squares the same, even on the diagonal. In this method, a 30-line pointed diagonally still covers 6 squares, and a 30-foot cone always covers 21 squares.

See Xanathar's Guide to Everything, page 86-87.

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