Pentagonal number

A pentagonal number, like square numbers and triangular numbers, is a type of figurate number. A figurate number is a number that can be represented using a regular geometric pattern typically formed using dots that are regularly spaced. A pentagonal number takes the form of a pentagon. The first 30 pentagonal numbers are:

1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335

Pentagonal numbers differ from square and triangular numbers in that the patterns formed are not rotationally symmetrical. Below is a figure showing the first 5 pentagonal numbers.

A pentagonal number represents the number of distinct dots that form the pentagons given that the number of dots that make up each side of the pentagon is equal to n, and the pentagons are overlaid such that they share one vertex. For example, the 3rd pentagonal number (P3) shown in the figure above has 3 dots per side and a total of 12 distinct dots.

Pentagonal numbers can be found using the following formula:


Find the pentagonal numbers for n = 1, 12, and 30.

1. n = 1:

2. n = 12:

3. n = 30:

How to determine if a number is pentagonal

To test whether a number is pentagonal, use the following formula, where x is a positive integer:

The positive integer, x, is a pentagonal number if and only if n is a natural number; in such a case, x is the nth pentagonal number.


Test whether 92 is a pentagonal number.

Since 8 is a natural number, 92 is the 8th pentagonal number, or P8 = 92.