Is pure 3-d (solid) geometry basically 2d?

Question

I recently got interested in "pure" geometry (i.e. that doesn't use equations with x,y,z). I did some fun 2-d stuff like a circle inscribed in a square with a smaller circle in the corner, then find the relative radii.
I then did similar 3-d stuff (a sphere in a cube, with a smaller sphere in the corner). In solving I noticed that I first converted to 2-d, by looking at a plane through the diagonal, then solved. So I got the impression than in pure solid geometry, you finally just end up resorting to 2-d geometry.
But I just watched Why slicing a cone gives an ellipse by 3Blue1Brown and they did a pure approach to a conic section. I loved it but, is that approach unusual? Or are there lots of 3-d problems that have a pure, non-2d, approach?