What is Volume in Math? Calculate Volume of Rectangular Prisms & Cubes w/ Units - [5-8-13] - By Math and Science
Transcript
00:00 | Hello . Welcome back . The title here is called | |
00:02 | Understanding Volume with pictures and models . This is part | |
00:06 | one . The idea in this lesson is to go | |
00:09 | from the concept of area which we learned in the | |
00:11 | last lesson to talk about one step beyond area which | |
00:15 | is called the volume of an object . So in | |
00:17 | this lesson we're going to concentrate on understanding what volume | |
00:20 | is in terms of pictures and models . And then | |
00:22 | later on we'll learn how to calculate volume . So | |
00:25 | in a nutshell , when you have the volume of | |
00:27 | an object , let's say you're talking about this three | |
00:30 | dimensional object , this is a cube here . We | |
00:32 | want to know how much space this cube occupies . | |
00:37 | How do we figure that out ? Remember for surface | |
00:39 | area let's say we're trying to find the area of | |
00:41 | this board . I would divide the board into little | |
00:44 | squares and so I would count the squares and it | |
00:46 | would be called square square centimeters or square millimeters and | |
00:50 | I would count them up . And that would be | |
00:51 | the area of this flat board . But what do | |
00:54 | I do when I have the volume of an object | |
00:56 | like this , instead of counting squares , I'm going | |
00:59 | to count cubes . It might be tough for you | |
01:00 | to see . But this model has little cubes all | |
01:03 | the way and you can see there's cubes here and | |
01:05 | there's cubes here and you can think of this large | |
01:07 | cube uh as being , how many cubes can I | |
01:11 | fit inside of this thing ? I would count them | |
01:13 | all . And that would tell me how large this | |
01:16 | thing is in terms of of cubic cubic centimeters or | |
01:20 | cubic inches or cubic miles or cubic yards or cubic | |
01:24 | light years or whatever . So instead of square inches | |
01:27 | for volume , we talk about cubic inches for area | |
01:31 | instead of square meters . We talk about in volume | |
01:34 | cubic meters . So it's the same concept . It's | |
01:36 | just that in a flat in a flat sheet to | |
01:39 | find area , we just count squares on the sheet | |
01:42 | . But when we have a three dimensional shape in | |
01:44 | space , how much space does it occupy ? We | |
01:46 | don't count squares . We count cubes . So it's | |
01:49 | called cubic meters or cubic centimeters and so on . | |
01:52 | So let's say for our first problem , we have | |
01:55 | a cube like this and we know that this side | |
01:58 | of the cube is uh well this is really not | |
02:01 | a cube . It's a it's a rectangle . All | |
02:03 | the dimensions are not the same size , but you | |
02:04 | get the idea this side this face of it is | |
02:06 | a cube three centimeters , three centimeters . And the | |
02:10 | depth of the cube are , The depth of the | |
02:12 | object is not quite three . It's a little bit | |
02:14 | shorter in the in the direction of the board , | |
02:16 | that's two centimeters . How do I measure how much | |
02:21 | space this thing occupies for that ? I'm gonna be | |
02:23 | talking about the volume . All right . So the | |
02:26 | first thing I want you to do is let's just | |
02:28 | look at the bottom layer of this thing . If | |
02:30 | I just cut everything off and look at the bottom | |
02:31 | layer , let's see what that would look like . | |
02:34 | All right . If I slice everything off , noticed | |
02:36 | this is still three centimeters and then going deep into | |
02:39 | the board , it's still two centimeters . Same dimensions | |
02:42 | is what I've had . All I've done is I've | |
02:43 | cut everything away and I've only only looking at the | |
02:46 | bottom layer . I can form a cube here . | |
02:50 | That is one centimeter in each direction and also one | |
02:53 | centimeter deep . We call it a cubic centimeter because | |
02:57 | just like a square centimeter is one centimeter on each | |
03:00 | side . A cubic centimeter is a cube , a | |
03:03 | little bit of cube . Like here , One centimeter | |
03:05 | in depth , one centimeter in width and one centimeter | |
03:08 | in height . One of these things is called a | |
03:10 | cubic centimeter . Just like in my little model here | |
03:14 | , one of these little cubes down here is called | |
03:16 | a would be called in that case a cubic centimeter | |
03:19 | as well . So if I slice it up and | |
03:21 | I have 123 cubes and then 456 cubes , I | |
03:25 | have six cubes on the bottom . So the volume | |
03:28 | of just the slice of the object is 123456 cubic | |
03:33 | centimeters . But the real object is all of that | |
03:36 | plus everything above it . So in order to find | |
03:39 | the volume of the whole object , we have to | |
03:41 | take this six and then we have to count everything | |
03:43 | that's above it as well . Which this is the | |
03:46 | original shape . You can compare this one to this | |
03:49 | one , It's the original shape . But now we | |
03:50 | have sliced cubes into the thing and we knew that | |
03:53 | there were six on the bottom layer . But now | |
03:56 | we have another layer of six and then we have | |
03:59 | another layer of six . So we have six plus | |
04:02 | six more cubes , plus six more cubes . So | |
04:05 | six times 3 . As another way to think about | |
04:07 | that is 18 . So if you wanted to find | |
04:10 | the volume of this thing , the volume V , | |
04:13 | it's going to be 18 and it's going to be | |
04:16 | cubic centimeters 18 cubic centimeters . Why is it 18 | |
04:23 | ? Because what we do is we take the shape | |
04:25 | . We knew that the shape was three centimeters tall | |
04:27 | and three centimeters wide and only two centimeters deep into | |
04:31 | the board . So we know that x centimeter a | |
04:35 | square here , since it's three tall , there has | |
04:37 | to be three of these little squares uh with one | |
04:40 | centimetre each going up and then one centimetre each going | |
04:43 | this way , and one centimetre each going this way | |
04:45 | . So when we take a look at it , | |
04:47 | we just start by looking at the base and we | |
04:49 | count six of those squares cubes , I'm sorry , | |
04:51 | six of those cubes . And then we have six | |
04:53 | more , that would be 12 and then six more | |
04:55 | for a total of 18 . So the volume of | |
04:58 | this object is 18 and the unit is not square | |
05:01 | centimeters . The unit for all volume is cubic centimeters | |
05:05 | or cubic meters or cubic millimeters or cubic yards or | |
05:09 | cubic miles or cubic light years . You get the | |
05:12 | idea it's always cubic because we're seeing how much space | |
05:15 | it takes up in a cube , little cubes that | |
05:17 | fit inside . Last thing I will tell you this | |
05:20 | lesson is mostly trying to get you to understand the | |
05:23 | concept 18 cubic centimeters . We can count them and | |
05:26 | we see that there's 18 cubes there but I want | |
05:28 | you to think for a second . Remember that area | |
05:31 | of a sheet is length times with right length times | |
05:34 | with what if I take three times 23 times two | |
05:39 | is six . That's the area of the bottom of | |
05:42 | the cube . When you think about it . I | |
05:43 | mean if this is a cube here and it was | |
05:45 | three times to then then you multiply them . That's | |
05:47 | the area of the bottom three times two or six | |
05:50 | . But if I multiply the six times the three | |
05:53 | here six times three , that's 18 , 18 . | |
05:56 | So the point of this lesson is not to calculate | |
05:59 | the area , I want to count the cubes . | |
06:01 | But I also want to tell you that to find | |
06:04 | the area of a sheet . It's just link times | |
06:06 | with to find the volume of some kind of cube | |
06:10 | or some kind of rectangular shape in three dimensional space | |
06:13 | . We call it a rectangular prism . The way | |
06:15 | you do it is you take length times width , | |
06:18 | times height , right length , times width , times | |
06:21 | height . So you can do it any order you | |
06:23 | want three times two or 66 times three is 18 | |
06:26 | . Let's go in another order three times 3 is | |
06:27 | nine . And then nine times two is also 18 | |
06:30 | . So to find the volume of any rectangular solid | |
06:33 | like that . You just do length , times width | |
06:36 | , times height . Now let me go take this | |
06:38 | down and get some more examples so that you can | |
06:39 | get practice . All right , Here's our next problem | |
06:43 | . We have a rectangular solid or also called a | |
06:45 | rectangular prison . That's two millimeters on the bottom by | |
06:49 | two millimeters deep . And then it's in height . | |
06:52 | It's four millimeters tall . How do we find the | |
06:54 | volume ? What we want to do is figure out | |
06:55 | how many cubes fit inside of this . But the | |
06:58 | units are in millimeters , so it's going to be | |
07:00 | cubic millimeters . How many cubes can I fit ? | |
07:03 | Where each cube is one millimeter on an edge . | |
07:06 | How many will fill inside this whole thing . So | |
07:08 | first we take a look at the bottom part . | |
07:10 | The bottom part remember was two millimeters wide and two | |
07:14 | millimeters deep . So if it's two millimeters wide , | |
07:17 | a cube one millimeter long , I'll fit two of | |
07:20 | them there . And if it's two millimeters deep , | |
07:22 | I'll fit to one millimeter uh cubes there . So | |
07:25 | you can see how many cubes will I have . | |
07:26 | I have 1234 cubes that are each way . This | |
07:30 | is one millimeter one millimeter one millimeter one millimeter one | |
07:34 | millimeter . So I have four of these cubes which | |
07:36 | are cubic millimeters . So I have four of them | |
07:39 | right here . And then if I look at the | |
07:40 | rest of the shape , I have four of them | |
07:43 | on the bottom layer , four of them here , | |
07:45 | four of them here and then four of them here | |
07:47 | . So it's 48 12 , 16 . Just count | |
07:52 | them 123456789 10 , 11 , 12 , 13 , | |
07:56 | 14 , 15 or 1314 , 15 , 16 . | |
08:00 | So there's 16 of them . So I would say | |
08:02 | the volume is 16 cubic with millimeters . So that | |
08:09 | means that there are 16 cubes that fit inside of | |
08:12 | the shape where each cube has one millimeter on each | |
08:15 | edge . Now I mentioned to you that to find | |
08:17 | the volume by calculation , you just take the length | |
08:20 | times the width times height . So two times two | |
08:23 | is four . And this four times to , I'm | |
08:25 | sorry , four times two is uh I'm sorry two | |
08:28 | times two is four and then four times four is | |
08:30 | 16 . So the formula holds but I want to | |
08:33 | stress in this lesson more how to count them than | |
08:35 | anything else . So let's take this one down and | |
08:37 | do another one . Alright . The next problem we | |
08:40 | have a rectangular solid like this with five m on | |
08:43 | this side and three m tall . And the depth | |
08:47 | into the board is four m . So how do | |
08:49 | we find the volume of this ? We want to | |
08:51 | find out how many cubic metres which are little cubes | |
08:54 | that each have a length of one m on each | |
08:56 | side . How many will fit in this ? So | |
08:58 | let's look First at the bottom layer five and four | |
09:01 | deep . So here again it's five wide and four | |
09:06 | deep . How many cubes are just in that layer | |
09:08 | ? What ? We have ? 12345 m and 1234 | |
09:12 | m . So we have , in terms of cubes | |
09:14 | 123456789 10 11 12 13 14 15 16 17 18 | |
09:21 | 1920 . So we have 20 cubes just in the | |
09:25 | bottom layer of this thing . Now let's look at | |
09:27 | the rest of the drawing and see how many more | |
09:29 | we have . We have 20 in the lowest layer | |
09:32 | . So we have 20 cubes in the lower layer | |
09:35 | , 20 more cubes in the middle layer and 20 | |
09:38 | more cubes in the top layer . So just like | |
09:40 | counting by twos , 246 you can count by 2020 | |
09:44 | . Then there's 40 than their 60 . So there's | |
09:48 | 60 cubes in that whole thing . So the volume | |
09:51 | is 60 cubic mhm meters . Mhm 60 cubic meters | |
09:59 | . Now let's see if , by calculating by multiplying | |
10:01 | it works five times four is 20 and then 20 | |
10:05 | times 3 2040 60 is 60 . So multiplying length | |
10:09 | times width , times height again also gives us the | |
10:12 | volume will take this down . And we have one | |
10:14 | more left in this lesson . All right , here's | |
10:16 | our last problem of this lesson . We have uh | |
10:20 | rectangular solid , that's uh two centimeters wide , three | |
10:24 | centimeters tall and four centimetres deep into the board . | |
10:28 | How many cubic centimeters are going to fit in ? | |
10:30 | Let's take a look at only the bottom layer . | |
10:33 | So if this was too wide and four deep , | |
10:35 | there must be uh two cubes each of one centimeter | |
10:38 | here and then 12344 going back each of one centimeter | |
10:43 | . So we have one cube , two cubes , | |
10:46 | three cubes , four cubes , five cubes , 678 | |
10:49 | cubes . So just on the bottom layer we have | |
10:52 | 88 cubes noticed . Two times four is eight . | |
10:55 | Okay and then how many layers do we have ? | |
10:58 | So if we have eight here and then ate here | |
11:00 | and then ate here eight times three is what ? | |
11:03 | Eight times three is 24 . So the volume is | |
11:06 | eight times 3 , 24 cubic centimetres , 24 cubic | |
11:13 | centimeters . And also remember that we can also calculated | |
11:15 | by multiplying two times four is eight and then eight | |
11:18 | times three is 24 cubic centimeters . So the point | |
11:21 | of this lesson really was to get you to understand | |
11:23 | what volume is just like area is just counting squares | |
11:28 | in a flat plane and how many squares fit on | |
11:30 | the surface . That's why we call the surface area | |
11:33 | volume is when we have a three dimensional shape . | |
11:35 | How many cubes ? Unit cubes cubic meters cubic centimeters | |
11:39 | will fit inside of an object . and just like | |
11:42 | area to find the area . You multiply length times | |
11:44 | with well to find the volume of a rectangular solid | |
11:47 | like this . You just link times width times height | |
11:50 | , right ? So you just have one more multiplication | |
11:51 | and you get the volume . So in this lesson | |
11:53 | , that's what we covered . I'd like you to | |
11:55 | solve these yourself . Follow me on the part two | |
11:56 | . We'll get a little more practice with understanding volume | |
11:59 | using models . |
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