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# Polygonal number

In mathematics, a polygonal number is a number that can be arranged as a regular polygon. Ancient mathematicians discovered that numbers could be arranged in certain ways when they were represented by pebbles or seeds. The number 10, for example, can be arranged as a triangle (see triangular number):

```   x
x x
x x x
x x x x
```
But 10 cannot be arranged as a square. The number 9, on the other hand, can be (see square number):

```x x x
x x x
x x x
```
Some numbers, like 36, can be arranged both as a square and as a triangle (see triangular square number):

```x x x x x x
x x x x x x
x x x x x x
x x x x x x
x x x x x x
x x x x x x
x
x x
x x x
x x x x
x x x x x
x x x x x x
x x x x x x x
x x x x x x x x
```
The method for enlarging the polygon to the next size is to extend two adjacent arms by one point and to then add the required extra sides between those points. In the following diagrams, each extra layer is shown as +.

Triangular numbers

1:

```+               x
```
3:

``` x               x
+ +             x x
```
6:

```  x               x
x x             x x
+ + +           x x x
```
10:

```   x               x
x x             x x
x x x           x x x
+ + + +         x x x x
```
Square numbers

1:

```+               x
```
4:

```x +             x x
+ +             x x
```
9:

```x x +           x x x
x x +           x x x
+ + +           x x x
```
16:

```x x x +         x x x x
x x x +         x x x x
x x x +         x x x x
+ + + +         x x x x
```
Polygons with higher numbers of sides, such as pentagons and hexagons, can also be represented as arrangements of dots (by convention 1 is the first polygonal number for any number of sides).

Pentagonal numbers:

1:

```+                   x
```
5:

``` x                   x
+ +                 x x
+ +                 x x
```
12:

```   x                   x
x x                 x x
+ x x +             x x x x
+     +             x     x
+  +  +             x  x  x
```
22:

```     x                   x
x x                 x x
x x x x             x x x x
+ x     x +         x x     x x
+ x  x  x +         x x  x  x x
+         +         x         x
+  +   +  +         x  x   x  x
```
35:

```       x                   x
x x                 x x
x x x x             x x x x
x x     x x         x x     x x
+ x x  x  x x +     x x x  x  x x x
+ x         x +     x x         x x
+ x  x   x  x +     x x  x   x  x x
+             +     x             x
+  +   +   +  +     x  x   x   x  x
```
Hexagonal numbers

1:

``` x
```
6:

```    x               x
+   +           x   x
+   +           x   x
+               x
```
15:

```     x                 x
x   x             x   x
+ x   x +         x x   x x
+   x   +         x   x   x
+       +         x       x
+   +             x   x
+                 x
```
28:

```        x                       x
x   x                   x   x
x x   x x               x x   x x
+ x   x   x +           x x   x   x x
+ x       x +           x x       x x
+   x   x   +           x   x   x   x
+     x     +           x     x     x
+       +               x       x
+   +                   x   x
+                       x
```
45:

```          x                           x
x   x                       x   x
x x   x x                   x x   x x
x x   x   x x               x x   x   x x
+ x x       x x +           x x x       x x x
+ x   x   x   x +           x x   x   x   x x
+ x     x     x +           x x     x     x x
+   x       x   +           x   x       x   x
+     x   x     +           x     x   x     x
+     x     +               x     x     x
+       +                   x       x
+   +                       x   x
+                           x
```
66: (which is also a triangular number and a sphenic number)

```          x                             x
x   x                         x   x
x x   x x                     x x   x x
x x   x   x x                 x x   x   x x
x x x       x x x             x x x       x x x
+ x x   x   x   x x +         x x x   x   x   x x x
+ x x     x     x x +         x x x     x     x x x
+ x   x       x   x +         x x   x       x   x x
+ x     x   x     x +         x x     x   x     x x
+   x     x     x   +         x   x     x     x   x
+     x       x     +         x     x       x     x
+     x   x     +             x     x   x     x
+     x     +                 x     x     x
+       +                     x       x
+   +                         x   x
+                             x
```
91:

```            x                             x
x   x                         x   x
x x   x x                     x x   x x
x x   x   x x                 x x   x   x x
x x x       x x x             x x x       x x x
x x x   x   x   x x x         x x x   x   x   x x x
+ x x x     x     x x x +     x x x x     x     x x x x
+ x x   x       x   x x +     x x x   x       x   x x x
+ x x     x   x     x x +     x x x     x   x     x x x
+ x   x     x     x   x +     x x   x     x     x   x x
+ x     x       x     x +     x x     x       x     x x
+   x     x   x     x   +     x   x     x   x     x   x
+     x     x     x     +     x     x     x     x     x
+     x       x     +         x     x       x     x
+     x   x     +             x     x   x     x
+     x     +                 x     x     x
+       +                     x       x
+   +                         x   x
+                             x
```
If s is the number of sides in a polygon, the formula for the nth s-polygonal number is ½n((s-2)n - (4-s)).

 Name Formula n=1 2 3 4 5 6 7 8 9 10 11 12 13 Triangular ½n(1n + 1) 1 3 6 10 15 21 28 36 45 55 66 78 91 Square ½n(2n - 0) 1 4 9 16 25 36 49 64 81 100 121 144 169 Pentagonal ½n(3n - 1) 1 5 12 22 35 51 70 92 117 145 176 210 247 Hexagonal ½n(4n - 2) 1 6 15 28 45 66 91 120 153 190 231 276 325 Heptagonal ½n(5n - 3) 1 7 18 34 55 81 112 148 189 235 286 342 403 Octagonal ½n(6n - 4) 1 8 21 40 65 96 133 176 225 280 341 408 481 Nonagonal ½n(7n - 5) 1 9 24 46 75 111 154 204 261 325 396 474 559 Decagonal ½n(8n - 6) 1 10 27 52 85 126 175 232 297 370 451 540 637 11-agonal ½n(9n - 7) 1 11 30 58 95 141 196 260 333 415 506 606 715 12-agonal ½n(10n - 8) 1 12 33 64 105 156 217 288 369 460 561 672 793 13-agonal ½n(11n - 9) 1 13 36 70 115 171 238 316 405 505 616 738 871 14-agonal ½n(12n - 10) 1 14 39 76 125 186 259 344 441 550 671 804 949 15-agonal ½n(13n - 11) 1 15 42 82 135 201 280 372 477 595 726 870 1027 16-agonal ½n(14n - 12) 1 16 45 88 145 216 301 400 513 640 781 936 1105 17-agonal ½n(15n - 13) 1 17 48 94 155 231 322 428 549 685 836 1002 1183 18-agonal ½n(16n - 14) 1 18 51 100 165 246 343 456 585 730 891 1068 1261 19-agonal ½n(17n - 15) 1 19 54 106 175 261 364 484 621 775 946 1134 1339 20-agonal ½n(18n - 16) 1 20 57 112 185 276 385 512 657 820 1001 1200 1417 21-agonal ½n(19n - 17) 1 21 60 118 195 291 406 540 693 865 1056 1266 1495 22-agonal ½n(20n - 18) 1 22 63 124 205 306 427 568 729 910 1111 1332 1573 23-agonal ½n(21n - 19) 1 23 66 130 215 321 448 596 765 955 1166 1398 1651 24-agonal ½n(22n - 20) 1 24 69 136 225 336 469 624 801 1000 1221 1464 1729 25-agonal ½n(23n - 21) 1 25 72 142 235 351 490 652 837 1045 1276 1530 1807 26-agonal ½n(24n - 22) 1 26 75 148 245 366 511 680 873 1090 1331 1596 1885 27-agonal ½n(25n - 23) 1 27 78 154 255 381 532 708 909 1135 1386 1662 1963 28-agonal ½n(26n - 24) 1 28 81 160 265 396 553 736 945 1180 1441 1728 2041 29-agonal ½n(27n - 25) 1 29 84 166 275 411 574 764 981 1225 1496 1794 2119 30-agonal ½n(28n - 26) 1 30 87 172 285 426 595 792 1017 1270 1551 1860 2197

References