No. You need to have the same vertex angle, or have the same ratio between the height and radius of the cones in order to have similar cones.
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A cone is 1/3 of the volume of a cylinder with the same base and height. A pyramid is 1/3 of the volume of a prism with the same base and height.
yes. they are always the same but a different scale.
There is no relationship, in the sense that you can have any base are with any height. For a prism shape (if the horizontal cross section is always the same as the base), the base area times the height is equal to the volume.
If they have the same nuber of sides (angles) then always. If not then never. So, overall, I suppose the answer is sometimes - depending on the number of sides.
No. Equiangular only implies the angles are the same, but says nothing about the lengths. Two equiangular triangles are always similar, only sometimes congruent.