 So section 12.1 is all about three-dimensional figures. You'll want to take some notes, so make sure you get this into your table of contents. We'll begin with some definitions. A polyhedron is a 3D solid in which each face is a polygon. Remember polygons have straight line segment sides, so this thing is a polyhedron. This thing is a polyhedron. Note the plural of the word polyhedron is polyhedra. So here I have two polyhedra. These three-dimensional figures are not polyhedra. The reason why is circles aren't polygons, and so those faces, if the faces are circles, they're not true polyhedra. They are 3D figures and we will study them, but just to make that distinction clear, the figures on the left are polyhedra, the figures on the right are not. So we have two basic types of polyhedra. We have our prisms and we have our pyramids. The distinction is how many bases there are. So in a prism, the bases are the two congruent parallel faces. So if you look at this prism, we have one, two, seven faces. The two faces that are congruent and parallel are this front pentagon and this back pentagon. And so that means this is a pentagonal prism. In a pyramid, the base is the one face that doesn't intersect any of the other faces. And so in this case, it's that pentagon on the bottom, and so here I have a pentagonal pyramid. So those are bases. We should also talk about lateral faces. Lateral faces are any faces that aren't bases. So in this pentagonal prism, the lateral faces are all of those rectangles. And so this pentagonal prism we see has five lateral faces. In a pyramid, the lateral faces are the isosceles triangles that sort of make up the sloped sides. So this pentagonal pyramid has five lateral faces. Some more definitions. An edge is that segment where two faces intersect. And a vertex is the point of the intersection for three or more edges. So I see this pentagonal prism. It has 15 edges and 10 vertices. Take a minute. Pause the video. How many edges and how many vertices does the pyramid have? We have 10 edges and 6 vertices. So there are an infinite number of other polyhedra. So for example, we have the icosa dodecahedron. We have the great cubic cube octahedron. There are an infinite number of other polyhedra. We're going to keep it rather simple for now. We'll get a bit more advanced as we move forward. But a few other polyhedra that we do need to talk about, the platonic solids. So there are five polyhedra that have congruent faces made up of regular polygons and the same number of faces meet at each vertex. There are only five polygons, sorry, only five polyhedra that do this. One is called the tetrahedron, and it's made up of four equilateral triangles. The second is called the hexahedron, which we more commonly known as a cube. It's made up of six squares. There's the octahedron, which is made up of eight equilateral triangles. The dodecahedron, which is made up of 12 regular pentagons. And the icosahedron, which is made up of 20 equilateral triangles. So these five make up the platonic solids. It's crazy, but there are no other possible platonic solids. There are only these five that meet these requirements of having congruent regular polygon faces. So finally, we've got a net, which is essentially an unfolding of a polyhedron. If we take a cube and unfold the box, we get something that kind of looks like an unfolded box. If I took this triangular prism, I could unfold it and kind of make this strange looking box. If I took this pentagonal pyramid, I could unfold it and I would see this kind of flower shape. And then finally, we'll do some isometric drawing practice, which is drawing in three dimensions. We'll actually use geojabra for these video notes. You'll use paper and pencil in class. So fire up geojabra and we'll get started. This video will use the isometric grid on geojabra. You'll use paper and pencil in class, but the mechanics of it are essentially the same. So let's start with this example. We want to draw a rectangular prism that's six units tall, two units long, and four units wide. So essentially, we've got a big box, a rectangular box. So my recommendation for you is start with the height. So I see I've got a prism that's six units tall. So I'm going to create a prism that's six units tall, not a prism, but I'm going to create a segment that's six units tall. So one, two, three, four, five, six. So there's a segment that's six units tall. Next, fill out one corner. And by that I mean I know I've got a rectangular lid or a base that's two units long, four units wide. So I'm going to create a rectangle with point B as one of its corners. So I know it's going to be two units long and one, two, three, four units wide. So B is kind of the front right corner of this lid, this box. And so now I'm going to kind of fill out the top. I know that C, B, and D represent three corners of a rectangular prism. So here would be the remainder of that base. So that is the top, that is the lid. We can even use the polygon tool to shade that in a little bit. And so even though angle C doesn't look like a 90-degree angle, it represents a 90-degree angle. Just like E and D and this angle C, B, D, they're representing 90-degree angles. So then from each of those top corners I'm going to draw down six, down six, down six. And then I'm going to fill out the bottom. So here would be the front right corner, front left corner. And now all that's left is I see this line would actually be sort of in the back hidden from view. And so I'll represent it as a dotted line. Likewise these two should be dotted. And now we're good to go. I think I might consider hiding these points, although I don't have to. If I turned off the grid you can sort of see the three dimensions pop out a little bit more. So there is our rectangular prism that's six units long and four units wide. Let's try another. Now let's draw a hexahedron with edges four units wide. So remember a hexahedron is another way of saying cube. So I'm going to start with the height, up four, four. And then I'm going to create the square top. So I've created the top, now I'm going to just draw down six. Sorry, draw down four, down four. And now I know three of these sides shouldn't be shown. And so I'll just represent them as dotted lines. And then maybe I'll shade the top. For the three dimensions to pop out a little bit better at you. So there we go, we've got our hexahedron, our cube, with edges four units. Instead let's hide the grid so you can see the three dimensions pop out a little bit better. So there is our hexahedron in isometric drawing view. And then one last example, let's draw a right triangular prism. It's two units tall and the legs are four by seven. So once again we're going to start with the height. Our height is two units tall, up two. And then we've got a right triangle as the two bases. So the legs are four and seven. So I'm going to go out four, four, and then out seven, seven. And now this is supposed to be a right triangle. So if I connect C to D, angle B here, C, B, D would represent a 90 degree angle. Even though it isn't, even though if you took a protractor to this it wouldn't be, it represents a right triangle. So then we'll draw down, back hypotenuse. It should be hidden from view. And then I'll use the polygon tool to sort of fill in the top. And so we've got our right triangular prism, two units tall with legs, four units, and seven units.