[size=150][b]The construction with ruler and compass of a decagon or a regular pentagon can be carried out by constructing the angle of 36° whose cosine is equal to[/b] [math]\frac{1}{\sqrt{5}-1}[/math].[/size][br]In this activity you can follow, by clicking on [color=#6aa84f][i][b]Start[/b][/i][/color] the steps to construct the 36° and 72° angles.[br][br][list][*]Let [color=#38761D][i][b]A[/b][/i][/color] and [i][b][color=#38761D]B[/color][/b][/i] two points whose distance [b][i][color=#38761D]AB[/color][/i][/b] is 1.[br][/*][/list][list][*]By drawing a [color=#38761D][b][i]circle [/i][/b][/color]with center [color=#38761D][i][b]B[/b][/i][/color] and radius [color=#38761D][i][b]AB[/b][/i][/color], we place a point [i][b]C[/b][/i] symmetrical to [color=#38761D][i][b]A[/b][/i][/color] in relation to [color=#38761D][i][b]B[/b][/i][/color].[br][/*][/list][list][*]Similarly, we place the point [i][b]D[/b][/i] symmetrical to [color=#38761D][i][b]B[/b][/i][/color] in relation to [i][b]C[/b][/i].[br][/*][/list][list][*]We raise the perpendicular to [color=#6aa84f][i][b](AB)[/b][/i][/color] [color=#38761D][color=#000000]to[/color][i][b] B[/b][/i][/color] noted [color=#3c78d8][i][b]By[/b][/i][/color] using the intersection of two circles of radius [i][b]AC[/b][/i] centred in [color=#38761D][i][b]A[/b][/i][/color] and [i][b]C[/b][/i]. Similarly, we draw the perpendicular to [color=#6aa84f][i][b](AB)[/b][/i][/color] in [color=#38761D][i][b][color=#000000][i][b]C[/b][/i][/color][/b][/i][/color] noted [color=#674ea7][i][b]Cy[/b][/i][/color] .[br][/*][/list][list][*]We place a point [color=#BF9000][i][b]E[/b][/i][/color] on [color=#674ea7][i][b]Cy[/b][/i][/color] such that [color=#BF9000][i][b]CE[/b][/i][/color]=[color=#38761D][i][b]AB[/b][/i][/color], it follows that [color=#BF9000][i][b]AE[/b][/i][/color]= [math]\sqrt{5}[/math] .[br][/*][/list][list][*]We draw a circle with center [color=#BF9000][i][b]E[/b][/i][/color] and with radius [color=#BF9000][i][b]CE[/b][/i][/color]. [br][/*][/list][list][*]This intersects the line [color=#BF9000][i][b](AE)[/b][/i][/color] at [color=#BF9000][i][b]F[/b][/i][/color].[br][/*][/list][list][*]The circle with center [color=#38761D][i][b]A[/b][/i][/color] and radius [color=#BF9000][i][b]AF [/b][/i][/color]intersects the line [color=#3c78d8][i][b]By [/b][/i]at[/color] [color=#38761D][i][b]G [/b][/i][color=#000000]and [color=#38761D][i][b]H[/b][/i][/color][/color][/color].[br][/*][/list][br]We have the following results : [color=#38761D][i][b]AG[/b][/i][/color]=[color=#BF9000][i][b]AF[/b][/i][/color]=[math]\sqrt{5}-1[/math] and cos([code][/code][color=#38761D][i][b]BÂG[/b][/i][/color])=[color=#38761D][i][b]AB[/b][/i][/color]/[color=#38761D][i][b]AG[/b][/i][/color]=[math]\frac{1}{\sqrt{5}-1}[/math] [br]then the angle [color=#38761D][i][b]BÂG[/b][/i][/color] is 36° and the angle [color=#38761D][i][b]GÂH[/b][/i][/color] is 72°.[br][br]From these angles we can construct a decagon and a regular pentagon.[br][br]