[b]The Goal:[/b][br]As the number of sides increses in polygons, drawing and counting diagonals becomes may be very difficult.[br]In this activity, you will discover a logic to calculate the number of diagonals using [color=#ff0000][b]Combinations[/b][/color], without counting them one by one.[br][br][b]Definitions:[/b][br][list][*][b]Vertices (n):[/b] The corners of the polygon.[br][/*][*][b]Edges (n):[/b] The outer lines that form the shape (black)[br][/*][*][b]Diagonals:[/b] The [u]inner lines[/u] connecting non-adjacent vertices. (red)[br][/*][/list][b]How to Explore:[/b][br][list=1][*]Use the [b]Slider [icon]/images/ggb/toolbar/mode_slider.png[/icon][/b] to change the number of vertices.[br][/*][*]Observe the lines connecting every corner to every other corner.[/*][/list]

[b]Step 1:[/b] The "Total Connections" Idea[br]Set the slider to any n you want. For example: n = 5 (Pentagon).[list][*]If we connect [b]every point to every other point[/b], how many lines do we get? [/*][/list][list][*]We are choosing 2 points out of 5 to draw a line. [b]That is the definition of [color=#ff0000]Combination![/color][/b][/*][/list]    [math]\binom{5}{2}[/math] = 10 lines in total.[br][br][b]Step 2: [/b]Separate the "Edges"[br]Look at the shape again (n=5).[br][list][*]The calculation above counts every line going out of a point.[br][/*][*][b]But we only want the Diagonals (the lines [i]inside[/i]).[/b][br][/*][*]Then we must remove the outer lines (the [b]Edges[/b]).[/*][/list][b]Step 3[/b]: The Formula[br]So, the logic is simple subtraction: [i]Number of Diagonals = Number of All Possible Lines - Number of Edges[/i][br][br]For a Pentagon (n=5):[br]Diagonals = 10 - 5 = 5[br][color=#cc0000][b]               [u]Conclusion (General Rule):[br][/u][/b][/color]             Number of Diagonals = [math]\binom{n}{2}-n[/math]