WOLFRAM|DEMONSTRATIONS PROJECT

Geometric Proof of the Tetrahedral Number Formula

​
blocks / edge
spacing
The tetrahedral numbers 1, 4, 10, 20, 35,
…
are the sums of the triangular numbers,
T
n
=1+3+6+10+…+
t
n
=(n+1)(n+2)(n+3)/6
.
This Demonstration justifies the formula by showing 6 tetrahedra with sides
n
forming an
(n+1)×(n+2)×(n+3)
cuboid. The controls allow you to vary the side length of the tetrahedra from 1 to 5 and to vary the spacing between them.