多边形公式的对角线

多边形可以定义为通过连接直线形成的封闭图形。所以很容易看出，要制作一个多边形，至少需要三条线。根据形成多边形的线数，已知的多边形有不同的名称。

多边形的对角线

多边形的对角线可以定义为连接任何多边形的两个相邻边的末端的线，它通常位于多边形内部。

示例：由 3 条线组成的多边形是三角形，由 4 条线组成的多边形是四边形，由 5 条线组成的多边形是五边形，依此类推。

多边形的对角线公式

Diagonals = (n × (n – 3))/2         

Where n is the number of sides of a polygon

证明

For making a diagonal in a polygon we need two vertices. Let’s consider an N-sided polygon, now each vertex can be connected to the other in ⁿC₂ different ways but in this, the number of sides which is n is taken twice so subtract n from the total number of ways.

Hence number of diagonals = ⁿC₂ – n

= (n!)/(n – 2)! × (2!) – n

= n(n – 1)(n – 2)!/(n – 2)! × (2!) – n

= n(n – 1)/2 – n

= (n(n – 1) – 2n)/2

= n(n – 3)/2

示例 1：三角形

取三角形并找出其中对角线的数量。

所以三角形有3条边所以n = 3

使用公式，对角线 = (n × (n – 3))/2

设 n =3

对角线 = (3 × (3 – 3))/2

= 0

因此，三角形的对角线为零。

示例 2：正方形

取 Square 并找出其中对角线的数量。

所以正方形有4条边所以n = 4

使用公式，对角线 = (n×(n-3))/2

把 n = 4

对角线 = (4 × (4 – 3))/2

= 2

因此，正方形有两条对角线。

示例 3：五角大楼

取五边形并找出其中对角线的数量。

所以五边形有5条边所以n = 5

使用公式，对角线 = (n × (n – 3))/2

把 n = 5

对角线 = (5 × (5 – 3))/2

= 5

因此，五边形有五个对角线。

示例问题

问题1：六边形有多少条对角线，用多边形公式求对角线。

解决方案：

Hexagon is a polygon that is formed by six straight lines

So a hexagon has 6 sides so n = 6

Using formula, diagonals = (n × (n – 3))/2  

Put n = 6

Diagonals = (6 × (6 – 3))/2  

= 9

Hence a hexagon has nine diagonals.

问题2：多边形有20条对角线，求它有多少条边？

解决方案：

Using diagonals formula = (n × (n – 3))/2  

So 20 = (n × (n – 3))/2  

20 × 2 = (n × (n – 3))

40 = n² – 3 × n

n² – 3 × n – 40 = 0 

n² – 8n + 5n – 40 =0

n(n – 8) + 5(n- 8) = 0

(n – 8)(n + 5) = 0

So n = 8

Hence the polygon is the octagon.

问题3：十边形有多少条对角线，用多边形公式的对角线求。

解决方案：

A decagon has 10 sides so n = 10

Using formula, diagonals = (n × (n – 3))/2  

Put n = 10

Diagonals = (10 × (10 – 3))/2  

= 35

Hence a decagon has 35 diagonals.

问题4：多边形有27条对角线，求它的边数？

解决方案：

Using diagonals formula = (n × (n – 3))/2  

So 27 = (n × (n – 3))/2  

27 × 2 = (n × (n – 3))

54 = n² – 3 × n

n² – 3 × n – 54 = 0

n² – 9n + 6n – 54 =0

n(n – 9) + 6(n- 9) = 0

(n – 9)(n + 6) = 0

So n = 9

Hence the polygon is Nonagon.

问题5：如果边是20，多边形有多少条对角线？

解决方案：

Put n = 20 in diagonals formula

Diagonals = (20 × (20 – 3))/2  

= 170

Hence there will be 170 diagonals in a 20 sided polygon.

问题6：多边形有405条对角线，求它有多少条边？

解决方案：

Using diagonals formula = (n × (n – 3))/2  

So 405 = (n × (n – 3))/2  

405 × 2 = (n × (n – 3))

810 = n² – 3 × n

n² – 3 × n – 810 = 0

n² – 30n + 27n – 810 =0

n(n – 30) + 27(n – 30) = 0

(n – 30)(n + 27) = 0

So n = 30

Hence the polygon has 30 sides.

问题 7：如果边数是 40，多边形有多少条对角线？

解决方案：

Put n = 40 in diagonals formula

Diagonals = (40 × (40 – 3))/2  

= 740

Hence there will be 740 diagonals in a 40 sided polygon.