using diagonals from a common vertex, how many triangles could be formed from the polygon pictured below?

The number of triangles is n−2n-2n−2, where nnn is the number of sides of the polygon.

Key idea

When you draw all diagonals from a single vertex of an nnn-sided polygon:

• That vertex is connected to every other vertex.

• The connections to the two adjacent vertices are just the sides of the polygon, not diagonals.

• So the number of meaningful diagonals from that vertex is n−3n-3n−3, and together with the two adjacent sides they partition the polygon into n −2n-2n−2 triangles (a “fan” of triangles around that common vertex).

How to use this for your picture

• Count how many sides the given polygon has (say it is a hexagon: n=6n=6n=6, a heptagon: n=7n=7n=7, etc.).

• Plug into the formula n−2n-2n−2:
  • If it is a hexagon, triangles =6−2=4=6-2=4=6−2=4.

* If it is a heptagon, triangles =7−2=5=7-2=5=7−2=5.

So: “Using diagonals from a common vertex, how many triangles could be formed from the polygon pictured below?”
Answer: count the sides nnn, then compute n−2n-2n−2.

Information gathered from public forums or data available on the internet and portrayed here.