Mathwords logoMathwords

Polygonal Number — Definition, Formula & Examples

A polygonal number is a number that can be represented by dots arranged in the shape of a regular polygon. Triangular numbers, square numbers, pentagonal numbers, and hexagonal numbers are all specific types of polygonal numbers.

The kk-th polygonal number with ss sides is defined as p(s,k)=k[(s−2)k−(s−4)]2p(s,k) = \frac{k\bigl[(s-2)k - (s-4)\bigr]}{2}, where s≥3s \geq 3 is the number of sides of the polygon and k≥1k \geq 1 is the index in the sequence.

Key Formula

p(s, k)=k[(s−2)k−(s−4)]2p(s,\,k) = \frac{k\bigl[(s-2)k - (s-4)\bigr]}{2}
Where:
  • ss = Number of sides of the polygon (s ≥ 3)
  • kk = Position in the sequence (k = 1, 2, 3, …)
  • p(s,k)p(s,k) = The k-th s-gonal number

How It Works

Each type of polygonal number corresponds to a regular polygon with ss sides. You build the sequence by nesting larger copies of that polygon around a central dot, adding a new layer each time. The general formula lets you compute any polygonal number from just two inputs: the number of sides ss and the position kk in the sequence. Setting s=3s = 3 gives triangular numbers, s=4s = 4 gives square numbers, s=5s = 5 gives pentagonal numbers, and so on.

Worked Example

Problem: Find the 4th pentagonal number using the general polygonal number formula.
Identify parameters: A pentagon has 5 sides, so s = 5. We want the 4th term, so k = 4.
s=5,k=4s = 5,\quad k = 4
Substitute into the formula: Plug the values into the general formula.
p(5,4)=4[(5−2)(4)−(5−4)]2=4[12−1]2p(5,4) = \frac{4\bigl[(5-2)(4) - (5-4)\bigr]}{2} = \frac{4\bigl[12 - 1\bigr]}{2}
Simplify: Evaluate the arithmetic inside the brackets and then divide.
=4×112=442=22= \frac{4 \times 11}{2} = \frac{44}{2} = 22
Answer: The 4th pentagonal number is 22.

Why It Matters

Polygonal numbers appear in combinatorics and number theory, including Fermat's polygonal number theorem, which states every positive integer is the sum of at most ss polygonal numbers of order ss. Recognizing the general formula helps you quickly derive or verify any specific figurate-number sequence in competition math or discrete mathematics courses.

Common Mistakes

Mistake: Using s=4s = 4 in the formula and expecting different results from k2k^2.
Correction: Setting s=4s = 4 correctly yields p(4,k)=k2p(4,k) = k^2. Square numbers are polygonal numbers—the formula must agree with familiar results, which serves as a good check.

Related Terms