median
don steward
mathematics teaching 10 ~ 16

## Tuesday, 11 March 2014

### lovely cuboids

cuboids with the same number (value) of the surface area and volume are interestingly different

there is a finite (smallish) number of cuboids with integer dimensions where surface area = volume

exploring the cube family,
the 6 by 6 by 6 cube is probably the simplest case (albeit a cube) that has the same value for both the volume and the surface area

let's refer to such cuboids as 'lovely'

the intention is that students work by trial and improvement, with some guidance (see the resources below)

but, to search for 'lovely' cuboids using algebra:
abc = 2bc + 2ac + 2ab
leads to the fraction relationship: 1/2 = 1/a + 1/b + 1/c
i.e. three unitary fractions that sum to a half

fix one of the variables:
'a' must be more than or equal to 3 and less than or equal to 6 (why?)
so there are restrictions, a = 3, 4, 5 or 6

with e.g. a = 3
b must be bigger than 6 (why?)
try 7, 8, 9, 10, ... and see if 'c' is an integer

there are 10 integer solutions altogether:
• five with a = 3
• three with a = 4
• one with a = 5
• and the cube with a = 6

the n, 6, 3 family of cuboids are also interesting
what is the difference between their surface area and volume values?
why?