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Current time:0:00Total duration:7:22

Measuring volume as area times length

Video transcript

I have this figure here you could call it a rectangular prism and I want to measure its volume and I'm defining my my unit cube as being a 1 centimeter by 1 centimeter by 1 centimeter cube has 1 centimeter with 1 centimeter depth 1 centimeter height and I will call this this is equal to 1 cubic cubic centimeters so I want to measure this volume in terms of cubic centimeters we've already seen that we can do that by saying hey how many of these cubic centimeters can fit into this figure without them overlapping in any way so if we had this you know in our hands we could kind of try to go around it and try to count it but it's hard to see here because there are some cubes that we can't see behind the ones that we are seeing so I'm going to try different tactics at it so first let's just think about what we can observe so we see that this one if we measure its different dimensions its width it's two of the unit length wide so it's two centimeters wide it's four of our unit length we're defining our unit length as a centimeter it's four of our unit length hi so this this dimension right over here is 4 centimeters and it is 3 of our unit length deep so this dimension right over here is 3 centimeters so I want to explore if we can somehow use these numbers to figure out how many of these cubic centimeters would fit into this figure and the first way I'm going to think about it is by looking at slices so I'm going to take I'm going to take a slice I'm going to take this slice right over here of our original figure and let's think about how we can how using these numbers we can figure out how many unit cubes were in that slice well this is 2 centimeters wide and it is 4 centimeters high it is 4 centimeters high and you might be saying hey Sal you know I could just count these things I could get 8 8 8 squares here but what if there was a ton there would be a lot harder and you might as well oh I could just multiply the width times the height that it would give me the area of this surface right over here and it's only 1 deep so that also will give me the number of cubes so let's do that let's find the area here that's going to be 2 centimeters times 4 centimeters 2 centimeters times 4 centimeters that gives us the area of this and then if we want to find out the number of cubes well that's also going to be equivalent to the number of cubes so we have 8 square centimeters is this area and the number of cubes is 8 and if we want the number of cubes in the whole thing we just have to multiply it by the number of slices and we see that we need 1 2 3 slices this is 3 centimeters deep so we're going to multiply that times 3 so we took the area of one surface we took the area of this surface right over here and then we multiplied it by the depth that essentially gives us the number of cubes because the area of this surface gives us the number of cubes in a 1 in a slice that is one cube deep and then we would have to have 3 slices like that so we would have to have this is one slice we would have to have another slice another slice and then another slice in order to construct the original figure so 2 centimeters times 4 centimeters times 3 centimeters would give us our volume let's see if that works out 2 times 4 is 8 times 3 is 24 24 let me do that in that pink color 24 centimeters cubed or I could say cubic centimeters so that's one way to measure the volume now there's multiple surfaces here I happen to pick this surface but I could have picked another one I could have picked this surface right over here and done the exact same thing so let's pick this surface and do the exact same thing this surface is 3 centimeters 3 centimeters by 4 centimeters by 4 centimeters let me do that in that blue color color changing is always difficult by 4 centimeters so it's area is going to be 12 square centimeters is the area of this surface and 12 is also the number of cubes that we have in that slice and so how many slices do we need like this in order to construct the original figure well we need it's 2 centimeters deep this is only one centimeter deep so we need two of them we need two of them to construct the original figure so we can essentially find the area of that first surface which was three times four three times four and then multiply that times times the width times how how many of those slices you need so times two and once again this is going to three times four is 12 times 2 is 24 I didn't write the unit's this first time but that's going to give us the count of how many cubic centimeters we have how many unit cubes we can fit so once again this is 24 cubic centimeters and you could imagine you could do the same thing not with this surface not with this surface but with the top surface the top surface is 3 centimeters deep 3 centimeters deep and 2 centimeters wide 2 centimeters wide so you could view its area or its area is going to be 3 centimeters times 2 centimeters so that area is let me do the same colors 3 centimeters times 2 centimeters 2 centimeters which is 6 square centimeters and that also tells you that there's going to be six cubes in this one cube deep slice but how many of these slices do you need well you have if this whole thing is 4 centimeters tall and this thing is only 1 centimeters you're going to need 4 of them so that's 2 2 3 3 try and draw it as neatly as I can 3 & 4 you're going to need four of these so to figure out the whole volume you have to take that and multiply that times 4 centimeters which once again 3 times 2 is 6 square centimeters times 4 centimeters is 24 24 cubic centimeters so any it doesn't matter what order you multiply these in you could view this take the ones area of one side then multiply it times the the depth we could take the surface fear of another height and multiply it times the the height or the width or the depth and these are all the scenarios but what it shows is it doesn't matter what what what order we multiply these three dimensions in you could view it you could take the 2 times the 4 first and then multiply by the 3 or you can take the three times the four first and then multiply by the two or you could take the two times the three first and then multiply by the four when you're multiplying it doesn't matter what order you're doing these in and so if you find if you have a rectangular prism like this and you know it's three dimensions you know it's two centimeters wide three centimeters deep and four centimeters tall you could say hey the volume of this thing the number of unit cubes the number of cubic centimeters it can fit is going to be two centimeters two centimeters times four centimeters times four centimeters times three centimeters which we've seen three times already is 24 cubic centimeters