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Polygonal Numbers

A polygon is a two-dimensional geometrical figure which is made by joining up straight lines together. In a regular polygon, all the sides and eventually all the angles are equal. Triangle (3 line segments), square (4 line segments), pentagon (5 line segments), hexagon (6 line segments), heptagon (7 line segments) etc are the examples of polygons. 

In mathematics, we often come across with the concept of polygonal numbers. It is a number which is represented in the form of dots arranged in such a way that it make a shape of regular polygon. Each of these dots are imagined as a unit. Polygonal numbers are a kind of two-dimensional figurate numbers.
For Example: 10 dots can be arranged in the shape of triangle as shown below :
Polygonal Numbers of 10

Thus, 10 is a triangle number. It cannot make a square. While, 9 dots can be easily arranged in square shape as below :
Polygonal Numbers of 9

Thus, 9 is a square number.

There are few numbers which can make a square as well as a triangle, such as 36 as demonstrated below:
Polygonal Numbers of 36

So, 36 is triangle number and square number both.
There are numbers which can be arranged in the form of polygons having higher numbers of sides. These numbers are known as pentagonal numbers, hexagonal numbers and so on.

Triangle numbers are shown below:
Triangular Numbers
Square numbers are demonstrated in the following figure:
Square Numbers
Pentagonal numbers are shown in the following diagram:
Pentagon Numbers

Hexagonal numbers are given in the figure below:
Hexagon Numbers

Let us go ahead and understand more about polygonal numbers in this article.

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Perimeter of a Polygon Calculator Number Rounding
 
The list of some polygonal numbers are given below:
Triangle Numbers:

1 , 3 , 6 , 10 , 15 , 21 , 28 ,  36 ,  45 ,  55 , ...

Square Numbers

1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 , ...

Pentagonal Numbers

1 , 5 , 12 , 22 , 35 , 51 ,  70 , 92 , 117 , 145 , ...

Hexagonal Numbers

1 , 6 , 15 , 28 , 45 , 66 , 91 , 120 , 153 , 190 , ...

Heptagonal Numbers

1 , 7 , 18 , 34 , 55 , 81 , 112 , 148 , 189 , 235 , ...

Octagonal Numbers

1 , 8 , 21 , 40 , 65 , 96 , 133 , 176 , 225 , 280 , ...

Nonagonal Numbers

1 , 9 , 24 , 46 , 75 , 111 , 154 , 204 , 261 , 325, ...

Decagonal Numbers

1 , 10 , 27 , 52 , 85 , 126 , 175 , 232 , 297 , 370 , ...

Formula

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The formulae of n$^{th}$ term of polygonal numbers are illustrated below:
Triangle Numbers

$T_{n}$ = $\frac{n}{2}$ (n + 1) = $\frac{n^{2} + n}{2}$  

Square Numbers

$T_{n}$ = $\frac{2n^{2} + 0.n}{2}$ =$n^{2}$

Pentagonal Numbers

$T_{n}$ = $\frac{3n^{2}-n}{2}$

Hexagonal Numbers

$T_{n}$ = $\frac{4n^{2}-2n}{2}$

Heptagonal Numbers

$T_{n}$ = $\frac{5n^{2}-3n}{2}$

Octagonal Numbers

$T_{n}$ = $\frac{6n^{2}-4n}{2}$

Nonagonal Numbers

$T_{n}$ = $\frac{7n^{2}-5n}{2}$

Decagonal Numbers

$T_{n}$ = $\frac{8n^{2}-6n}{2}$

and so on.

We see that we have to remember the pattern that is followed in each formula. Here, in each successive formula after triangle numbers, the coefficient of $n^{2}$ has to be increased by 1, also the coefficient of n is decreased by 1. Thus, we need not to byheart them. 
The sum of the first n consecutive polygonal numbers can also be found. There is a particular pattern which is followed by the formulae of sum of these numbers, so that the students need not cram all the formulae.

These formulae are given below:

Triangle Numbers

$S_{n}$ = $\frac{1}{6}$ $n (n+1) (n+2)$

Square Numbers

Now, in each successive formula, everything remains same except for the terms in second bracket. Here, coefficient of n is increased by 1 as well as constant term in decreased by 1.

$S_{n}$ = $\frac{1}{6}$ $n (n+1) (2n+1)$

Pentagonal Numbers

$S_{n}$ = $\frac{1}{6}$ $n (n+1) (3n+0)$ = $\frac{1}{2}$ $n^{2} (n+1)$

Hexagonal Numbers

$S_{n}$ = $\frac{1}{6}$ $n (n+1) (4n-1)$

Heptagonal Numbers

$S_{n}$ = $\frac{1}{6}$ $n (n+1) (5n-2)$

Octagonal Numbers

$S_{n}$ = $\frac{1}{6}$ $n (n+1) (6n-3)$ = $\frac{1}{2}$ $n (n+1) (2n-1)$

Nonagonal Numbers

$S_{n}$ = $\frac{1}{6}$ $n (n+1) (7n-4)$

Decagonal Numbers

$S_{n}$ = $\frac{1}{6}$ $n (n+1) (8n-5)$

and so on.

Examples

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Some solved examples based on the concept of polygonal numbers are illustrated below.

Example 1: Find the 20$^{th}$ pentagonal number as well as the sum first 20 pentagonal numbers.

Solution: The formula of n$^{th}$ pentagonal number is -

$T_{n}$ = $\frac{3n^{2}-n}{2}$

Putting n = 20

$T_{20}$ = $\frac{3 \times 400-20}{2}$

= $\frac{1200-20}{2}$

= $\frac{1180}{2}$ = 590

The formula of sum of first n pentagonal number is -

$S_{n}$ = $\frac{1}{2}$ $n^{2} (n+1)$

Putting n = 20

$S_{20}$ = $\frac{1}{2}$  $\times$  $400 (21)$

= 200 x 21

= 4200

Example 2: Calculate the sum of first 8 square numbers.

Solution: The formula for finding sum of n square number is -

$S_{n}$ = $\frac{1}{6}$ $n (n+1) (2n+1)$

Substituting n = 8

$S_{8}$ = $\frac{1}{6}$ $\times$ $8(8+1) (2 \times 8+1)$

= $\frac{1}{6}$ $\times 8(9) (17)$

= 4(3) (17)

= 204

Example 3: Calculate the value of 10$^{th}$ hexagonal and heptagonal numbers.

Solution: The formula for n$^{th}$ hexagonal number is -

$T_{n}$ = $\frac{4n^{2}-2n}{2}$

Put n = 10

$T_{n}$ = $\frac{4 \times 100-2 \times 10}{2}$

= $\frac{400-20}{2}$

= $\frac{380}{2}$

= 190

The formula for n$^{th}$ heptagonal number is -

$T_{n}$ = $\frac{5n^{2}-3n}{2}$

Put n = 10

$T_{n}$ = $\frac{5 \times 100-3 \times 10}{2}$

= $\frac{500-30}{2}$

= $\frac{470}{2}$

= 235
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