Return to Yes, 3D

Triangular Numbers

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, …

Triangular numbers

Triangular numbers are a beautiful thing. It is a visual pattern, a numeric pattern, and there is a nice algebraic function. The is the mathematical trifecta we love.


My introduction to triangular numbers (at the 1:12 minute mark, to 4:58)

Video by a school student on triangular numbers

Introduction to tetrahedral numbers, which are made up of triangles (at the 6:49 mark).


The triangular numbers are useful.

They answer the Handshake Question: If 12 people are in a room and each person shakes hands with every other person once, how many handshakes are there?  A: T_{11} = 66 handshakes.

This can be used anytime you have to find out how many pairings there are. For example for a recreation volleyball or softball league.

T_{n} is the number of sides and diagonals of an n-sided polygon.

The formula (function) to find the nth triangular number is  T_{n}=\frac{n(n+1)}{2} 


Triangular Number Sequence – from MathIsFun.

Fascinating Triangular Numbers

Wiki on triangular numbers

Tetrahedral numbers

Wiki on tetrahedral numbers