In class my group of three each tackled one of the shapes, and we ended up making them so the cube was made. But, I (selfishly) wanted a set of my own because I was so intrigued by these little things. So, I got out my fancy paper and drew, and cut, and pasted away for a couple hours and this was the result.
First things first, I needed a plan. I'm not one to just "wing it" so I got out a piece of paper and sketched what I thought would be the correct nets without looking at the work from class. Here is my sketches and notes. I knew I wanted my cube to be 2 in. by 2 in. (which ended up being quite lucky because my paper was just big enough for a 2" by 2" cube).
However, as I worked with the nets, I wanted to find what I thought would be the most efficient net, or essentially the most compact. I also wanted to make sure all of my square root measurements were as exact as I could get them by drawing those parts attached to a right triangle with that length hypotenuse (for example, by lines that were to be the square root of 8 inches long I drew as a hypotenuse to a 2" by 2" triangle. Therefore, some of the nets I ended up using deviated from the original plan. However, I thought of this as a learning experience not only for myself, but others who make these types of figures as well. It got me thinking about how many unique nets could be made that all produce the same figure.
In class we discussed how we knew that these figures would work to make a cube. We discussed to volume of each shape and concluded that the triangular prism composed 1/2 the volume of the cube since the formula is 1/2(w*l*h). The square pyramid constituted 1/3 of the volume of the whole cube since the formula is 1/3(w*l*h), and that the triangular pyramid makes up 1/6 of the volume since the formula would be (1/3)*(1/2)*(l*w*h). Basically, as long as the smaller figures have a combined volume that is equal to the larger figure and the correct design they will fit together to form the larger figure.
I decided first to make the full cube. This way I could reference the cube as needed for the construction of the three more difficult pieces I was about to construct. (Remember when I said I was lucky that I chose 2" by 2"? This is where I discovered that - the cube net wouldn't have fit had it been much bigger!) Below is the net I used for the cube. I strategically placed it in such a way that I utilized the longest length of paper I had in order for the whole cube it fit. It tuned out well!
(I ended up having to cut off one of the tabs shown in the picture of the net because I had one too many)