Pyramidal number

Key takeaways

Pyramidal number belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Pyramidal number to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Pyramidal number from memory before moving on to harder problems.

Reference excerpt

A pyramidal number is the number of points in a pyramid with a polygonal base and triangular sides. The term often refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to a pyramid with any number of sides. The numbers of points in the base and in layers parallel to the base are given by polygonal numbers of the given number of sides, while the numbers of points in each triangular side is given by a triangular number. It is possible to extend the pyramidal numbers to higher dimensions.

Formula The formula for the nth r-gonal pyramidal number is

P n r = 3 n 2 + n 3 ( r − 2 ) − n ( r − 5 ) 6 , {\displaystyle P_{n}^{r}={\frac {3n^{2}+n^{3}(r-2)-n(r-5)}{6}},}

where r ∈ N {\displaystyle r\in \mathbb {N} } , r ≥ 3. This formula can be factored:

P n r = n ( n + 1 ) ( n ( r − 2 ) − ( r − 5 ) ) ( 2 ) ( 3 ) = ( n ( n + 1 ) 2 ) ( n ( r − 2 ) − ( r − 5 ) 3 ) = T n ⋅ n ( r − 2 ) − ( r − 5 ) 3 , {\displaystyle P_{n}^{r}={\frac {n(n+1){\bigl (}n(r-2)-(r-5){\bigr )}}{(2)(3)}}=\left({\frac {n(n+1)}{2}}\right)\left({\frac {n(r-2)-(r-5)}{3}}\right)=T_{n}\cdot {\frac {n(r-2)-(r-5)}{3}},}

where Tn is the nth triangular number.

Sequences The first few triangular pyramidal numbers (equivalently, tetrahedral numbers) are:

1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ... (sequence A000292 in the OEIS) The first few square pyramidal numbers are:

1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, ... (sequence A000330 in the OEIS). The first few pentagonal pyramidal numbers are:

1, 6, 18, 40, 75, 126, 196, 288, 405, 550, 726, 936, 1183, ... (sequence A002411 in the OEIS). The first few hexagonal pyramidal numbers are:

1, 7, 22, 50, 95, 161, 252, 372, 525, 715, 946, 1222, 1547, 1925 (sequence A002412 in the OEIS). The first few heptagonal pyramidal numbers are:

1, 8, 26, 60, 115, 196, 308, 456, 645, 880, 1166, 1508, 1911, ... (sequence A002413 in the OEIS)

References

Illustrations

Worked examples

Example 1 — a first encounter with Pyramidal number

Start with the simplest possible case. Write down what Pyramidal number claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Pyramidal number before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Pyramidal number ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Pyramidal number

In research: Pyramidal number appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Pyramidal number in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Pyramidal number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Figurate numbers, so understanding it makes those chapters shorter.
In everyday life: Look for Pyramidal number outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Pyramidal number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Pyramidal number means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Pyramidal number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pyramidal number in simple terms?

A pyramidal number is the number of points in a pyramid with a polygonal base and triangular sides. The term often refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to a pyramid with any number of sides.

Why does Pyramidal number matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Pyramidal number?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Pyramidal number.