Figurate Numbers

Figurate numbers are sequences generated by figures made up of evenly spaced dots. Figurate numbers include the triangular numbers, square numbers, pentagonal numbers, etc. Sometimes they are called polygonal numbers since they form polygons.


Triangular Numbers
1
3
6
10
15
 
 














The sequence of triangular numbers is generated by dots making right triangles of increasing size. This sequence 1, 3, 6, 10, 15, ... has the general formula of n(n+1)/2 where n is the ordinal number of the figure, like the first, second, third triangle picture.

So the sixth triangle picture will have n = 6 in the formula, or 6(6+1)/2 = 21 dots. How many dots will the 15th triangle have?


Square Numbers
1
4
9
16
25

 

 

 














The sequence of square numbers is generated by dots making squares of increasing size. This sequence of 1, 4, 9, 16, 25, ... is the sequence of perfect square numbers. That's where they got their name, from the square figures that these special numbers make.

What is the general formula that generates these numbers? How many dots will the 10th square have?


More on polygonal numbers
Pentagonal numbers
Hexagonal numbers
Heptagonal numbers
Octogonal numbers
Triangular numbers and Pascal's Triangle