There were a number of partial solutions but this well explained one is almost entirely the work ofAndrei of Tudor Vianu National College and shows how "obvious" the answer is with a little visualising and careful reasoning.


Plan of stand


Let l be the exterior length of the cylinders and r - the radius of the cylinders.

In the first solution, I have 4 cylinders (the 4 squares length l-4r and 2r) and 8 half-cylinders - cylinders cut through the diagonal (the 8 small right-angled triangles from the figure).

Each of the four cylinders has the volume: V1 = pr2 (l - 4r)

Each of the half-cylinders has a volume of half a cylinder:
V2 = 1
2
 ×pr2 ×2r = pr3

Now, the total volume of the tubular stand is: V = 4 ×V1 + 8 ×V2 = 4pr2 (l - 2r)

With the second method, I arrange the 8 half-cylinders to make 4 bigger cylinders, and these 4 to the bigger ones. The height of one 'big' cylinder is (l - 2r), and the total volume is: V = 4 pr2 (l - 2r)

which is exactly the result obtained above. Substituting the numerical values: l = 10 cm and r = .5 cm, I obtain:

V = 4 p×0.25 ×(10 - 1) = 9 p = 28.27 cm3

If the volume wood would be double (18 p), then the outside dimension of the dowel would be:
18 p
=
4p×0.25 (l -1)
l - 1
=
18 cm
l
=
19 cm
If the volume of wood would be the same but the radius would be 1 cm, then the outside dimension would be:
9 p
=
4p(l - 2)
l - 2
=
9
4
l
=
4.25 cm
The general formula for the volume of the dowel (proved above) is:

V = 4 pr2 (l - 2r)

One could see that the volume is proportional to the square of the radius, and in the limit of long outside dimensions, proportional to the outside dimension.