I understand that to work out the number of faces of a connected planar graph, you use Euler's formula F = A - N + 2, where A is the number of arcs and N is the number of nodes.

For a triangle node (3 arcs and 3 nodes), the number of faces would therefore be 3 - 3 + 2 = 2.

But I can't count two faces, only one (the triangle itself). Where is the second face, or where have I misunderstood?

up vote 3 down vote accepted

You probably smack your head against the wall for this but: One face "inside" the triangle, the other one is "outside".

  • 2
    And this is true for all planar graphs -- it's not just about triangles. – David Richerby Apr 5 at 18:46

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