Figurate Number — Definition, Formula & Examples

A figurate number is a number that can be represented by a regular arrangement of dots in a geometric shape, such as a triangle, square, or pentagon. The most common examples are triangular numbers (1, 3, 6, 10, ...) and square numbers (1, 4, 9, 16, ...).

Figurate numbers are members of sequences defined by arranging discrete units into regular polygonal or polyhedral patterns. For a polygon with $s$ sides, the $n$ th $s$ -gonal figurate number is given by $P(s, n) = \frac{n[(s-2)n - (s-4)]}{2}$ , where $n \geq 1$ .

Key Formula

P(s,\, n) = \frac{n\,[(s-2)\,n - (s-4)]}{2}

Where:

$s$ = Number of sides of the polygon (3 for triangle, 4 for square, etc.)
$n$ = Term number in the sequence (1st, 2nd, 3rd, ...)
$P(s, n)$ = The nth s-gonal figurate number

How It Works

To build a figurate number, start with a single dot and progressively add layers that form the chosen shape. For triangular numbers, each new row adds one more dot than the previous row: 1, then 2, then 3, and so on. For square numbers, each new layer wraps around the previous square by adding the next odd number. The general polygon formula lets you compute figurate numbers for pentagons, hexagons, and beyond by plugging in the number of sides

s

and the term number

n

Worked Example

Problem: Find the 5th triangular number using the figurate number formula.

Identify values: A triangle has 3 sides, so

s = 3

. We want the 5th term, so

n = 5

s = 3, \quad n = 5

Substitute into the formula: Plug

s = 3

and

n = 5

into the general formula.

P(3, 5) = \frac{5\,[(3-2)(5) - (3-4)]}{2} = \frac{5\,[5 - (-1)]}{2} = \frac{5 \times 6}{2}

Calculate: Simplify to get the answer.

P(3, 5) = \frac{30}{2} = 15

Answer: The 5th triangular number is 15. You can verify this: 1 + 2 + 3 + 4 + 5 = 15.

Why It Matters

Figurate numbers appear throughout number theory, combinatorics, and even physics (e.g., stacking cannonballs). Recognizing that the sum

1 + 2 + 3 + \cdots + n

equals a triangular number helps you quickly compute series sums, a skill used frequently in algebra and competition math.

Common Mistakes

Mistake: Assuming all figurate numbers are just perfect squares.

Correction: Square numbers are only one type. Triangular, pentagonal, and hexagonal numbers are all figurate numbers too, each with a different geometric pattern and formula.