Question

How many diagonals does a twenty sided polygon have?

Answer 1

See: https://www.mathsisfun.com/geometry/polygons-diagonals.html

Explanation:

The number of diagonals of an n-sided polygon is:

`(n(n − 3))/ 2`

Substituting `20` for `n` gives:

`(20(20 - 3))/2 = 10(20 - 3) = 10 * 17 = 170`

Answer 2

Any diagonal is drawn between two vertices. A 20-sided polygon has 20 vertices.

The diagonals can be counted as `AB`, `AC`, `AD`, …, `BC`, `BD`, `BE`, …, `ST`, the combination of choosing `2` points from `20` vertices, or `""_20C_2=((20),(2))`. However, this method also counts the sides as diagonals. Remove the `20` sides to get `((20),(2))-20=190-20=170`.

In general, the diagonals for a convex `n`-sided polygon is `((n),(2))-n=(n!)/(2!*(n-2)!)-n=(n(n-1))/2-n=(n(n-3))/2`.