Difference of Cubes

y

opacity

figure

boxes

frustums 1

frustums 2

3

x

-

3

y

= 3×

1

3

(x-y) (

2

x

+xy+

2

y

) = (x-y)(

2

x

+xy+

2

y

)

If a cube is put into the corner of a larger cube, the region between them can be split up into three boxes of the same width. That region can also be split up into three congruent truncated pyramids (frustums). If the inside cube is put into the middle of the larger cube, the region between them can be split up into six congruent square frustums.

The volume of a frustum is

1

3

h(A+

AB

+B)

, where

h

is the height and

A

and

B

are the areas of the base and top.