# A Visual Proof of Nicomachus's Theorem

A Visual Proof of Nicomachus's Theorem

Nicomachus's theorem states that ++…+=, where is a positive integer. In words, the sum of the cubes from 1 to is equal to the square of the sum from 1 to .

3

1

3

2

3

n

2

(1+2+…+n)

n

n

n

For a visual proof, calculate the total area in the figure in two different ways: First, count the unit squares from the center to an edge to get , so that the total area is . Second, consider that each square ring consists of squares of side , with area .

1+2+3+...+n

4

2

(1+2+...+n)

4k

k

4

3

k