Pyramidal Number — Definition, Formula & Examples

A pyramidal number is a figurate number that represents the total count of objects arranged in a pyramid with a polygonal base, where each layer is a smaller polygon stacked on top of the previous one.

The $k$ th square pyramidal number is defined as the sum of the first $k$ perfect squares: $P_k = \sum_{i=1}^{k} i^2$ . More generally, pyramidal numbers arise from summing successive polygonal numbers of a given type, producing a three-dimensional figurate number.

Key Formula

P_k = \frac{k(k+1)(2k+1)}{6}

Where:

$P_k$ = The kth square pyramidal number
$k$ = The number of layers in the pyramid (a positive integer)

How It Works

To find a pyramidal number, you stack layers of polygonal numbers. For a square pyramid, the bottom layer has

k^2

objects, the next has

(k-1)^2

, and so on up to

1^2

at the top. You sum all the layers to get the total. The closed-form formula lets you skip the summation and compute the result directly.

Worked Example

Problem: Find the 5th square pyramidal number.

Step 1: Write out the sum of the first 5 perfect squares.

P_5 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25

Step 2: Alternatively, apply the closed-form formula with k = 5.

P_5 = \frac{5 \cdot 6 \cdot 11}{6} = \frac{330}{6} = 55

Answer: The 5th square pyramidal number is 55.

Visualization

Why It Matters

Pyramidal numbers appear in combinatorics—for instance,

P_k = \binom{k+2}{3} + \binom{k+1}{3}

connects them to binomial coefficients. They also show up in physics when counting stacked objects (like cannonballs) and in computer science when analyzing nested loop iterations.

Common Mistakes

Mistake: Confusing square pyramidal numbers with tetrahedral numbers.

Correction: Tetrahedral numbers sum triangular numbers (

1, 3, 6, 10, \ldots

), while square pyramidal numbers sum perfect squares (

1, 4, 9, 16, \ldots

). The formulas differ: tetrahedral uses

\frac{k(k+1)(k+2)}{6}

, square pyramidal uses

\frac{k(k+1)(2k+1)}{6}