613=6eh+5.196152e^2

Question

Surface area of a hexagonal prism is 613 mm^2. Find the dimensions (h is height, e is edge length).

Answer 1

Surface area of hexagon = 613=6eh + 3e²√3. 3√3=5.196152 approx.

Therefore h=(613-3e²√3)/(6e).

There is no unique solution but since h is positive, e must be less than 10.86mm approx.

Here are some related values.

e (mm)	h (mm)
0.5	203.90
1	101.30
1.5	66.81
2	49.35
2.5	38.70
3	31.46
3.5	26.16
4	22.08
4.5	18.81
5	16.10
5.5	13.81
6	11.83
6.5	10.09
7	8.53
7.5	7.13
8	5.84
8.5	4.66
9	3.56
9.5	2.53
10	1.56
10.5	0.64

When the height and edge are equal h=e=7.3994mm approx.