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.