A k-gonal number is a type of figurate number that represents a polygon with k sides. These numbers can be visualized as dots arranged in the shape of a k-sided polygon. For example, triangular numbers are 3-gonal numbers, while square numbers are 4-gonal numbers. The formula to calculate the nth k-gonal number is given by P(k, n) = \fracn((k-2)n - (k-4))2 . K-gonal numbers can be used to explore patterns in mathematics and number theory. They extend beyond simple shapes, allowing for the study of relationships between different types of polygons. Understanding k-gonal numbers can also lead to insights in areas such as combinatorics and algebra.