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# Side Lengths

We’re now on the worksheet on page two, we’re going to do the volumes and surface areas. But first, we must work out the length of each of the sides in our model of tonic solids, and we’re going to do that in column D. I’ve assumed that the cube is one and just to make things easier from there, we have to work out the side of the tetrahedron. tetrahedron sits in the cube like that. Can you work out the length of the song The tetrahedron Of course. It’s the hypotenuse of a triangle with a and b sides of one and one. So that’s right to. The next one we look at is the octahedron which sits inside the tetrahedron like that. Can you work out the length All the sides of the octahedron Well, it has each of the sides and each of those octahedron sides is a ecological triangle hits root two over two and the next is the icosahedron which sits inside the octahedron like that happens to sit in a golden ratio position. How does this divide the sides of the octahedron? Firstly, because that will give us the key to working out the side of the icosahedron. Let us assume that the octahedron has a side of one That would mean that the site would be divided like this into one over five and one over five squared. Now, if we were to take one triangle, a right triangle at the corner, it would have shorter sides of one over five squared and one over five squared. Then using ecosys theorem, that will give us the length of the side. But we assumed that the octahedron had a side length of one, when in fact it doesn’t, we know it has a side length of root two over two, we must multiply our answer by root two over two. And that turns out to be point 381966, and so on. But that equals one over five squared. So in our model, the length of the side of the icosahedron is one over five squared. That leaves only the dodecahedron and the cube and scribes inside the dodecahedron like this. We know the length of the side of the cube is one. What are the sides of the dodecahedron That side of the cube is the diagonal of a regular pentagon. If that diagonal equals one, then the sides are one Iver fi and one over fi dodecahedron as a side length of one over five. We’ve now completed the length of all the sides. We can now go ahead and work out the volumes and surface areas. There’s something interesting to note about the length of the sides. We know that one over five squared plus one over five equals one That means that in our model, the side length of the icosahedron plus the side length of the dodecahedron equals the side length of the cube. Well, that’s something interesting to note. Now we know the length of the sides of each of the solids in our model, we’re ready to go on and study the volumes. Now for this, you’ll need formulas, but I’ve written all those on the worksheet. So all you have to do is go through and substitute the value of each of the sides into the formula and then we’ll see what we get