The other day in math class my professor showed us a paper cube he brought from home, however this was no ordinary cube. It was held together by magnets and was able to split into three different parts. The cube was split in half into two equal right triangular prisms. One of the prisms was then split once more, into a square pyramid with its vertex directly above one of its corners, and its height equal to the height of the original cube, and a third piece I can only think to describe as an irregular tetrahedron. We were split into groups to reconstruct the three polyhedra starting with their nets. I thought this was a fun exercise, and thought I would take it further outside of class.
I began with the task of constructing a specific cone, i.e. one with given dimensions. I decided I would specify the radius of the base, r, and the angle, θ, to be measured between one end of a diameter, the vertex, and the opposite end of the diameter. I first drew a qualitative net of a cone, and labeled the parts I would need to find. In my case, I needed to find the radius, R, of the pie shaped portion of the net (in the picture below, its one mighty piece of pie), as well as an angle, Φ, which must be spanned by the piece so the arc length equals the circumference of the base. I began by drawing a cross section of the cone through a diameter and the vertex. This was an isosceles triangle which I split into two congruent right triangles. I used simple trigonometry to find R in terms of r and phi. I found that R = r/sin(θ/2). I then found Φ in terms of given quantities. The key to this was simply to realize that the arc spanned by Φ must equal the circumference of the base. The formal representation of this is ΦR = 2πr, which after substituting in our value for R and solving for Φ, I concluded that Φ = 2πsin(θ/2). To test this I chose θ = 90° and a non specific radius (it ended up being 3.04 “Geogebra cms”), and used Geogebra to create my net. After folding, sure enough the angle was (about) 90°!
Feeling that the quantitative aspect cones and pyramids would reduce to trig problems, I decided to do some qualitative constructions too. I started with the problem of constructing a square pyramid whose vertex was directly above the midpoint of one of its edges, for the sake of explanation let it be the right edge. I came to the conclusion that such a net consists of two different isosceles triangles, two congruent right triangles, and one square. The right side (where the vertex lies) is an isosceles triangle with its height equal to the height of the pyramid, and its base equal to the edge of the square base. The front and back side are two congruent triangles with one leg equal to the edge of the square base, and another equal to the equal legs of the isosceles triangle on the right. The left side is the second isosceles triangle, and has two equal legs that are equal to the hypotenuse of the right triangles previously described. I again used Geogebra to make the net and construct the solid.
Wanting yet a little more, I turned my attention to constructing a net for a cylinder with a whole in the middle. I could easily construct a net for a cylinder with holes in its bases, but figuring out how to connect to that the inner surface necessary to make our cylinder a solid was a bit trickier. I started by drawing two cylinders, one large (with holes in its bases), and one small. For the smaller cylinder, I used dotted lines to represent the base, as the “base” of the small cylinder is the hole in the base of the large cylinder. I tried to superimpose the two images (matching the “base” of the small cylinder to the hole in the big one), but could not. This was due to the overlap of the base of the large cylinder with the side of the small cylinder. I was ready to claim that the net was impossible to construct when the idea of splitting one of the large bases in half hit me. This would allow me to attach the side of the small cylinder to its “base” (the hole in the big base) without forcing any overlap of the big base and the small side. I found out the hard way that in order for the base splitting to work, one must be careful to attach the halves at opposite sides of the side. This is not opposite ends on the net, as the two ends come together at the same point. After a failed attempt, and one minor change, I was able to construct the net for a cylinder with a hole in it.
Throughout the process I learned that when working with nets it is very helpful to first do a qualitative analysis before a quantitative one. I also found that a good method was to start where things are simple, i.e. a base or known side, and stem from there. I would be interested to work out the quantitative description of the cylinder with a hole removed from its center. I also think it could be easily generalized in a way that the hole and cylinder need not be coaxial.