Many thanks to Andrei Lazanu of School 205 Bucharest for the inspiration for this solution. Well done Andrei. One might have expected the triangular solution to be best. I wonder what happens if you chop off the corners?

Using the following notation:

h - for the height of the cup and

d - for the diameter of the cup.

Test each of the two possible forms of the box, one with a rectangular base and one with a triangle as its base:

Cuboidal box

As there are six cups they could be ordered in two ways:

1 x 6 cups

and

2 x 3 cups


For any cuboid box, the surface area is: 2 times the top area + 2 times the side area + 2 times the front area:

The total area of a 1 x 6 cups box is:

2 ×(h ×6 ×d + d ×6 ×d + d ×h) = 185470 mm2

The total area of a 2 x 3 cups box is:

2 ×(h ×2 ×d + h ×3 ×d + 2 ×d ×3 ×d) = 157250 mm2


For a triangular prism box
This case is illustrated in the figure of the problem.
Diagram of equilateral triangle

The base is an equilateral triangle. The surface area is 2 times the area of the equilateral triangle + 3 times the area of each of the faces.


First calculate the side of the equilateral triangle ABC.

This is 2 x BM + 2 x 85~mm.

BM is a tangent to the circumference of a corner cup (centre O, radius d/2 = 85/2~mm).

Also the angle MOB = 60°

Therefore angle MBO =30°.


BM
=
 85
2
 ×tan 60°
=
 85
2
 ×√3
=
 85√3
2
The side of triangle ABC is
=
2 ×d + 2 ×BM
=
2 ×85 + 2 ×  85√3
2
=
85 ( 2 + √3 )
The altitude of the equilateral triangle ABC is
=
AB ×sin 60°
=
85(2+√3) ×  √3
2
=
 85√3 (2+√3)
2
Area of the two equilateral triangles
=
85 ( 2 + √3 ) ×  85√3 (2+√3)
2
=
87149  mm2
Area of the three faces
=
3 ×83 ×85 ( 2 + √3 )
=
78989  mm2
The total surface area is
=
166138 mm2

So the best box is a box of 2 x 3 cups.