Today in class we spent our time trying to create nets that mapped our the figures out professor had. These figures were a triangular prism, a triangular pyramid, and a square pyramid that were all shaped in such a way that they fit together to form a cube. The tricky part here was that we weren't just making an prism or pyramid, but they had to fit together just right to form the cube.
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!
Next on my to do list was to build the triangular pyramid. I had the task of building this figure in class, so I already knew one way to do it (see the plan). However, I wanted to stretch myself to see how efficient I could make the net for it, an I must admit I'm pretty proud of this one! Look at that beautifully simply net! I started off by drawing a right triangle with the two legs length 2". I did this because I knew that this piece constituted one half of two of the faces of the cube, hence I knew I would need two triangles this size. Next, I knew that the other two triangles would also have a length of 2" on on side in order to fit against the triangle I just drew. I thought that the best way to do this would be to draw a rectangle connected to the hypotenuse of the triangle with a width of 2". I then divided this triangle in half to produce those two triangles I needed. The last step was to draw another 2" by 2" right triangle on the opposite side. In order to draw this triangle I had to draw 45 degree angles off both ends of the rectangle I drew to ensure that the third angle would be 90 degrees. The resulting net and constructed figure are shown below.
(I ended up having to cut off one of the tabs shown in the picture of the net because I had one too many)
Onto the next figure! I constructed the triangular prism next because I thought it would be the easiest to make. First I made the full 2" by 2" face as it was the easiest starting point. Then I connected another face of the cube. I knew I needed two right triangles with leg length 2" to connect the two faces, so that was next. Then the last part-the diagonal that spans the middle of the competed cube. Using the Pythagorean Theorem I knew the length needed to be the square root of 8 and the easiest way to make that was to draw the rectangle attached to the hypotenuse of one of the triangles. I could have drawn this connected directly to one of the faces, but thought it would be harder to measure out the square root of 8 and therefore less efficient. The results are below.
Finally I had just one more figure to make, and this one gave me the most trouble- the square pyramid. I knew how we had drawn it in class, and that is what is seen on the original plan. But I thought, "If I draw the same size triangles together it should be easier, right?" Wrong. I drew out the whole net like in the picture below, and started to make the tabs to use when I folded it up. That is where I realized where I went wrong. There was no way that those triangles on the top left and top right (ignore the one with the dotted lines-those were to help me get the correct length) were going to connect to the face in the middle of the net. So I erased it and went back to square one.
I ended up drawing the net the way I had originally planned as I couldn't find a different net that produced the same figure. This time I again started with the square face in the middle. I knew that I needed two 2" by 2" right triangles to make two halves of two other faces. Next, I knew I needed to make two other triangles that when folded would meet the other two I just drew. These ended up being the same size as the triangles I made in the middle rectangle when I constructed the triangular pyramid earlier. The results are below.
The time had finally come. Did I make all these the right way? Would they fit together? Scroll through the slideshow below to see all the individual pieces and then all of them put together.
It Worked!
I really enjoyed making all of these nets today and it really made me think about what all goes into making these nets so they fit together in this way. Not only do they have to be compatible shapes, but also have the right angles and sizes. Had I not seen the models in class today I would have struggled a lot to try to figure out how to get all the shapes to fit together so nicely. I also thought about how much knowledge is really necessary to make these. I think really you could get by by just knowing he basic shapes and the original side length for most of these. For example, in the triangular prism, I knew that the rectangle I made had to be the square root of 8 inches long, but it wasn't really necessary. A younger student could analyze the figure are realize they needed a rectangle there that was as long as the hypotenuse of the triangle and 2" long and just drew the rectangle without ever knowing its length. This is one reason I like doing these. They seem very versatile. They can be used very basically for just building shapes, or they can be looked at very analytically if you wish to compare the relationships of the figures to build something bigger (like the cube) out of them.
An afterthought....
After making these 3D figures, I started to wonder what other 3D figures I could make using all three of these as part of it. This thought began when I arranged the figures like in the picture below. I thought, "I could just make a big rectangular prism with these if I made some more pieces that fit on top of these." But, alas, my paper wasn't long enough and neither was my patience. Though I do think it would be rather cool to have a whole set of these that fit together to form lots of different figures. Maybe I will tackle that idea someday.
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