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human beings have always realized that certain things are longer than other things for example this line segment looks longer than this line segment but that's not so satisfying just to make that comparison you want to be able to measure you want to be able to quantify how much longer the second one is then the first one and how do we go about doing that well we define a unit length we define a unit length so if we make this our unit length this we say this is one unit then we could say how many of those lengths are each of these lines so this first line looks like it is we can do one of those units and then we could do it again so it looks like this is two units while this third one looks like we can get to see that's one to three of the units so this is three of the units and right here I'm just saying units sometimes we've we've made conventions to define a centimeter where the unit might look something like this and it's going to look different depending on your screen or we might have an inch that looks something like this or we have might have a foot that I won't be able to fit on this on the screen based on how big I've just drawn the inch or meter so there's different units that you could use to measure in terms of but now let's think about more dimensions this is literally a one dimensional case this is 1d why is it one dimension well I can only measure measure I can only measure length but now let's go to a 2d case let's go to two dimensions where objects could have a length and a width or width and a height so let's imagine two figures here that look like this so let's say this is one of them this is one of them and notice it has a width and it has a height or you could view it as a length in it or you could view it as a width and a length depending how you want to view it so let's say this is one figure right over here and let's say this is the other one so this is the other one right over here try to draw them reasonably well now once again now we're in two dimensions and we want to say well how much how much in two dimensions space is this taking up or how much area are each of these two taking up well once again we could just make a comparison this second if you viewed the miss carpets or rectangles the second rectangle is taking up more of my screen than this first one but I want to be able to measure it so how would we measure it well once again we would define a unit square instead of just a unit length we now have two dimensions we have to define a unit square and so we might make our unit square and the unit square we will define as being a square where its width and its height are both equal to the unit length so this is its width is one unit and its height is one unit and so we will often call this one square unit oftentimes you'll say this is one unit and you put this two up here this literally means one unit squared and instead of writing unit I could have this could have been a centimeter so this would be one square centimeter but now we can use this to measure these areas and you say well how many just as we said how many of this unit length could fit on these lines we could say how many of these unit squares can fit in here and so here we might take one of our units squared and say okay it fills up that much space or we need more to cover all of it well there we'll put another unit square there we'll put another unit square right over there we'll put another unit square right over there Wow for unit squares exactly cover this so we would say that this has an area of four four units four square units or four units squared now what about this one right over here well here let's see I could fit one two three four five six seven eight and nine so here I could fit nine nine units nine units squared let's keep going we have we live in a three-dimensional world why restrict ourselves to only one or two so let's go to the 3-d case and once again when people say 3d they're talking about three dimensions they're talking about the different directions that you can measure things in here there's only length here there's length and width Thor width and height and here there will be width and height and depth so once again if you have if you have let's say a an object and now we're in three dimensions where in the world we live in that looks like this and then you have another object that looks like it looks like this that looks like this it looks like this second object takes up more space more physical space than this first object does it looks like it has a larger volume but how do we actually measure that and remember volume is just how much space something takes up in three dimensions area is how much space something takes up in two dimensions length is how much space something takes up in one dimensions but when we think about space we're normally thinking about three dimensions so how much space would you take up up take up in the world that we live in so just like we did before we can define instead of a unit length or unit area we can define a unit volume or a unit cube so let's do that let's define our unit cube and here it's a cube so all of its its length width and height are going to be the same value so my best attempt at drawing a cube and they're all going to be one unit so it's going to be one unit high one unit one unit deep and one unit one unit wide and so to measure volume because a well how many of these unit cubes can fit into these different shapes well this one right over here and you won't be able to actually see all of them I could essentially break it down into so let me see how well I can do this so that we can count them all it's a little bit harder to see them all because you're there's some cubes that are behind us but if you think of it as two layers so one layer would look like this one layer is going to look like this so imagine two things like this stacked on top of each other so this one's going to have one two three four cubes now this is going to have two of these stacked on top of each other so here you have eight unit cubes or you could have eight units cubed volume what about here well if we try to fit it all in if we try to fit it all in let me see how well I could draw this well I can draw this it's going to look something like this it's going to look something like this obviously this is kind of a rough drawing and so if we were to try to take this apart you would essentially have three you'd have a stack of three sections that will each look something like this my best attempt at drawing it three sections that would look look something like what I'm about to draw so it would look like this so three if you took three of these and stacked them on top of each other you would get this right over here and each of these have one two three four five six seven eight nine cubes in it nine times three you're going to have twenty-seven twenty-seven cubic units in this one right over here so hopefully that helps us think a little bit about how we measure things especially how we measure different things in different number of dimensions especially in three dimensions when we call it volume