Question

What is the total number of diagonals for a hexagon and a heptagon?

Answer 1

The previous answer correctly gave the formula for a number of diagonals `D` in `N`-sided convex polygon:
`D = (N(N-3))/2`
Below is its explanation.

Explanation:

Let's fix one particular vertex in a convex polygon. It has two neighboring vertices that are connected to our vertex by two polygon's sides. All other `N-3` vertices can be connected to our vertex by a diagonal.

So, from each vertex we can draw `N-3` diagonals to other vertices in a convex polygon.

Now we have to take into consideration that this can be done at each of `N` vertices. So, the number of diagonals seems to be `N(N-3)`. However, in this process we counted each diagonal twice. If `A` and `B` are two vertices that can be connected by a diagonal, we counted one and the same diagonal twice - firstly, as on of those initiated from `A` and, secondly, as one of those initiated from `B`.

Therefore, the real number of diagonals is half of what we counted above, that is
`D = (N(N-3))/2`

With this formula in mind, for hexagon (`6`-sided convex polygon, that is `N=6`) the number of diagonals is
`D_6 = (6*3)/2 = 9`

For `N=7`
`D_7 = (7*4)/2 = 14`