In order to calculate the electric flux passing through one side of a cone with no net charge enclosed, I originally thought you needed to take infinitesimal areas and dot the normal vector with the field vector and integrate. However, I found that you can take the 2D projection of the cone, find the area, and multiply it by the field to get flux. However, I'm a bit confused on why this works. The area of the 2D projection is different from the surface area of one half the cone, and Gauss's law has an integral of areas. 

I have a feeling that the discrepancy of area is made up by the fact that the dot product between the vectors makes up for this difference. Does this have something to do with how vector projection ties into the dot product? I'm not super familiar with the deep intuition of it, but I'm curious why this discrepancy in area gives the same flux.