The question that led Euler to his investigations on graphs is the well-known question related to the seven bridges of Königsberg, and that story is a must in every introduction to graph theory.

My question is whether it is also known what led Euler to investigate on polyhedra; was it a 'recreational' question, or was it a mathematical research question of his contemporaries?

A further question would be whether Euler continued his investigations on polyhedra after having discovered his famous characterisation of polyhedra of genus 0.