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# Ways to cross-section a cube

CCSS.Math:

## Video transcript

what I want to do in this video is explore the types of two-dimensional shapes we can construct by taking planar slices of cubes so what am I talking about well let's say we wanted to construct a square how could we slice a cube with a plane to get a the intersection of this cube and that plane to be a square well imagine if that plane were to cut just like this a square is maybe the most obvious one so it cuts the top right over there it cuts the top right over there it cuts this side right over here cuts this side right over here it cuts the side I guess in the back if it's a glass cube you'd be able to see it right over there dotted line and then it cuts this right over here so you could imagine you could imagine a plane that did this and if I wanted to draw the broader plane I could draw it like this let me see if I can do a decent an adequate job of drawing the actual I guess you could say part of the plane that is cutting this cube that is cutting this cube it would look it could look something it could look something like this and I can even color in the part of the plane that you could actually that you could actually see if the cube were if the cube were opaque if it were if you couldn't see through it and actually but if you could see it through it you see this dotted line and the plane would look like that so a square is pretty is a pretty straightforward thing to get if you're doing a planar slice of a cube but what about a rectangle how can you get that and at any point I encourage you pause the video and try to think about it on your own how can you get these shapes that I'm talking about well a rectangle you can actually cut like this so if you cut this side like this and then cut that side like that and then you cut this side like that I think you see where this going this side like that and then you cut the bottom right over there then the intersection of your the plane that you're cutting with so the intersection let's see this could be the plane that I'm actually cutting with so the intersection of the plane that I'm cutting with and my cube is going to be is going to be a rectangle so it might look like this and once again let me shade in the stuff if you kind of view this this if you if you imagine the plane is like one of those those huge blades that magicians use to solve people in half or pretend like they're or give us the illusion of sawing people in half it might look something like this okay so you're like okay I you know that's not so hard to digest that I can if I intersect a plane with a cube I can get a square I could get a rectangle but what about triangles well once again and pause the video if you think you can figure it out triangles not so bad you could cut this side right over here this side this side right over here and this side right over here and then this is and of course I could keep drawing the plane but I think you get the idea this would be a triangle there's different types of triangles that you can construct you could construct a you can construct an equilateral triangle so as long as this cut is the same length as this cut right over here is the same length as hot weird Enders or the length that it intersects on this space of the cube that's going to be an equilateral triangle if you pushed this point out more actually let me do that in a different color if you push it out more you're going to have an isosceles triangle you're going to have an isosceles triangle if you were to bring this point really really really close like here you would approach having a right angle but it wouldn't be it wouldn't be quite a right angle you'd still have these angles would still be less than 90 degrees you can approach 90 degrees so you can't quite have an exactly a right angle and so you can't get to 90 degrees you definitely not going to get to 91 degree so you're actually not going to be able to do an obtuse triangle either but you can do an equilateral you can do an isosceles you can do scalene triangles I guess you could say you could do the different types of acute triangles but now let's do some really interesting things can you get a Pentagon by slicing a cube with a plane and I really want you to pause the video and think about it here because that's just just a fun to think think about how can you get a pentagon by slicing a cube with a plane alright so here I go this is how you can get a Pentagon by slicing a cube with the plane imagine slicing the top let me do it a little bit different so imagine slicing the top right over there so we imagine slicing the top like this but imagine slicing this backside imagine slicing this backside like that this backside that you can't see quite like that now you slice this side right over here like this like this and then you slice this side right over here like this this could be like if I wanted to draw on the plane maybe it won't be so obvious if I try to draw the plane but you get the actual idea if I slice this the right angle not at a right angle at the right at a right angle actually I should you zoom in with a right angle that'll confuse everything if I slice it in the proper angle that I am with the intersection of my plane and my cube is going to be is going to be this Pentagon right over here now let's up the stakes something let's up the stakes even more what about a hexagon can I slice a cube in a way with a two-dimensional plane to get at the intersection of the plane and the cube being a hexagon well as you could imagine I wouldn't have asked you the question unless I could so let's see if we can do it so if we slice this if we slice this right over there if we slice this bottom piece right over there and then you slice this back side like that you slice this backside like that and then you slice this side that we can see right over there and then this side that we can see right over there to the side I could have written it much straighter so hopefully you get the idea I could I can slice this cube so that I can actually get a hexagon so hopefully this this gives you a better appreciation for what you can actually do with a cube especially if you're busy slicing it with a large planer Planes large planer blades in in some way there's actually more to a cube than maybe you might have imagined in the past and one way to think about it is there are six sides to a cube and so are six surfaces to a cube so you can cut as many as six of the surfaces when you intersect it with a plane and every time you cut into one of those surfaces it forms a side so here we're cutting it to four sides here we're cutting it to four surfaces or four sides here we're cutting into three here cutting into five we're not cutting into the bottom of the cube here and here we're cutting into all six of the sides all six of the surfaces of the faces of this cube