Math Antics - Volume - By Mathantics
Transcript
00:03 | Uh huh . Hi , I'm rob . Welcome to | |
00:07 | Math Antics . In this lesson we're going to learn | |
00:09 | about another important geometry quantity called volume . Specifically we're | |
00:13 | gonna learn what volume is and what kind of units | |
00:15 | we use to measure volume and how we can calculate | |
00:18 | the volumes of a few simple geometric shapes . The | |
00:21 | first thing you need to know is that volume is | |
00:23 | a quantity that all three dimensional objects have . But | |
00:26 | to understand what it means , it will help if | |
00:28 | we back up just a little and start out with | |
00:30 | a one dimensional object like this line segment to measure | |
00:34 | a one dimensional object , we need a one dimensional | |
00:36 | quantity which we usually call length . The length of | |
00:39 | this line happens to be exactly one centimeter which is | |
00:42 | a common unit for measuring length . And in the | |
00:45 | Math Antics video about area . We saw that if | |
00:48 | we move or extend this one dimensional line in a | |
00:51 | direction perpendicular to it by a distance of one centimeter | |
00:54 | . It forms a two dimensional object called the square | |
00:57 | . Two dimensional objects are measured by the two dimensional | |
00:59 | quantity that we call area because the original line was | |
01:03 | one centimeter long and we extended it a distance of | |
01:06 | one centimeter . The amount of area that this square | |
01:09 | occupies is one square centimeter , which is a common | |
01:12 | unit for measuring area . Okay , now imagine that | |
01:16 | we take that two dimensional square and move or extend | |
01:19 | it in a direction perpendicular to its surface by a | |
01:22 | distance of one centimeter . It forms a three dimensional | |
01:25 | object that's called a Cube . And to measure a | |
01:28 | three dimensional object like this , we use a three | |
01:30 | dimensional quantity called volume volume , tells us how much | |
01:34 | three dimensional space or three D . Space and object | |
01:36 | occupies . All right , but how much volume does | |
01:40 | this cube have ? Well , since it was made | |
01:42 | by extending one square centimeter a distance of one centimeter | |
01:45 | in the third dimension , we say that its volume | |
01:48 | is exactly one cc which is a common unit for | |
01:51 | measuring volume . So square units are used to measure | |
01:54 | area and cubic units are used to measure volume . | |
01:58 | Since square units are made by multiplying too one dimensional | |
02:01 | units together like centimeter time centimeter . We can abbreviate | |
02:04 | them using the exponent notation centimeters to the second power | |
02:08 | or centimeters squared . And since cubic units are made | |
02:12 | by multiplying 31 dimensional units together like centimeter time centimeter | |
02:16 | time centimeter . We can abbreviate them using exponent notation | |
02:20 | centimeters to the third power or centimeters cubed . And | |
02:25 | just like there are different sizes of square units like | |
02:28 | square inches square meters or square miles . There's also | |
02:32 | different sized cubic units like cubic inches , cubic meters | |
02:35 | or cubic miles . See how area and square units | |
02:39 | are related to volume and cubic units . And there's | |
02:42 | another similarity to , do you remember how you can | |
02:45 | use square units to measure the area of any two | |
02:48 | D shape ? Not just squares . Well you can | |
02:51 | use cubic units to measure the volume of any three | |
02:53 | D shape . Not just cubes to see how that | |
02:56 | works . Take a look at this two D . | |
02:57 | Circle and this three D object called a sphere , | |
03:00 | which is like a ball just like you can use | |
03:03 | a bunch of small squares to approximate the area of | |
03:06 | the circle . You can use a bunch of small | |
03:08 | cubes to approximate the volume of the sphere . And | |
03:11 | here's the really cool part . The smaller the squares | |
03:14 | you use , the closer their combined area will match | |
03:16 | the area of the circle and the smaller the cubes | |
03:19 | you used , the closer their combined volume will match | |
03:22 | the volume of the sphere . Okay , so volume | |
03:26 | is a three D quantity for measuring three D objects | |
03:29 | . But since we've also talked a lot about area | |
03:30 | in this video , I want to point out that | |
03:32 | three D objects also have a type of area that | |
03:35 | you don't want to confuse with volume . You might | |
03:38 | remember from our previous videos that two D shapes have | |
03:40 | both a two D quantity called area and a one | |
03:43 | D quantity called perimeter . Well , in a similar | |
03:46 | way , three D objects have both a three D | |
03:49 | quantity called volume and a two D quantity called surface | |
03:52 | area . Surface area is a lot like it sounds | |
03:56 | . It's the area of the objects . Outer surface | |
03:58 | or shell perimeter and surface area are both outer boundaries | |
04:03 | of geometric shapes . Perimeter is the one dimensional outer | |
04:06 | boundary of a two dimensional shape . And surface area | |
04:09 | is the two dimensional outer boundary of a three dimensional | |
04:12 | shape . And to help you see the difference between | |
04:15 | surface area and volume , Imagine that you have a | |
04:17 | perfectly thin box filled with ice . If you unfold | |
04:21 | the box you can see the two D area that | |
04:23 | surrounds the volume . Well , the volume itself is | |
04:26 | the amount of three D . Space occupied by the | |
04:28 | ice inside the box . All right . So now | |
04:31 | that you understand what volume is and you won't confuse | |
04:33 | it with surface area , we're going to spend the | |
04:35 | rest of the video learning how to calculate the volumes | |
04:37 | of some basic geometric shapes . But before we do | |
04:40 | that , I want to quickly mention something important about | |
04:43 | terminology where the words we use to describe things in | |
04:45 | math . Most of the time people agree on what | |
04:48 | to call things in math . But not always . | |
04:50 | That's especially true when it comes to the words that | |
04:53 | we use to describe the dimensions of geometric objects . | |
04:56 | Take the words length , width and height . For | |
04:58 | example , if I have a rectangle , I could | |
05:00 | name it's two dimensions , length and width like we | |
05:03 | did in the area video but I could also name | |
05:05 | them width and height if I wanted to . The | |
05:07 | actual names of the dimensions really are important . So | |
05:10 | different teachers might use different names . The important thing | |
05:14 | is to be flexible and realize that the math concepts | |
05:16 | are the same even if different words are used to | |
05:18 | explain them . For example , the area of a | |
05:21 | rectangle is always found by multiplying its two side dimensions | |
05:24 | together no matter what they're called . Okay , back | |
05:28 | to calculating volumes , a lot of three D shapes | |
05:31 | can be formed by taking a two D . Shape | |
05:33 | and then extending it along the third dimension . For | |
05:36 | example , if you start with a rectangle and then | |
05:39 | extend it along the third dimension . You get a | |
05:41 | three D . Shape called a rectangular prison . If | |
05:45 | you start with a triangle and then extend it along | |
05:47 | the third dimension . You get a three D . | |
05:49 | Shape called a triangular prism . And if you start | |
05:52 | with a circle and extend it along the third dimension | |
05:54 | , you get a three D . Shape called a | |
05:56 | cylinder . From the others . You might have thought | |
05:58 | that it should be called a circular prison . But | |
06:01 | technically prisons are shapes that are formed by extending a | |
06:04 | polygon . And since the circle is not a polygon | |
06:06 | the resulting shape is not called a prison . Okay | |
06:10 | so the good news is that there is a general | |
06:12 | formula for calculating the volume of these types of three | |
06:15 | D shapes . All you have to do is find | |
06:17 | the area of the original two D . Shape they | |
06:19 | got extended and then multiply that by the length or | |
06:22 | distance that it was extended . Usually that original shape | |
06:26 | is called the base of the object . So let's | |
06:29 | start with this rectangular prism and calculate its volume . | |
06:32 | The base is the original rectangle and its dimensions are | |
06:35 | four cm x three cm . So to find the | |
06:38 | area of that base , we need to remember how | |
06:40 | to calculate the area of a rectangle . To do | |
06:42 | that . We just multiply the two side dimensions together | |
06:45 | four centimeters times three centimeters gives us 12 centimeters squared | |
06:49 | as the area . Now to find the volume . | |
06:52 | We just need to multiply that area by the length | |
06:55 | that the base was extended . Our diagram tells us | |
06:58 | that distance is 10 centimeters . So if we multiply | |
07:01 | 12 centimeters squared by 10 centimeters we get 120 centimeters | |
07:05 | cubed . So this rectangular prism has a volume of | |
07:08 | 120 cubic cm . Okay let's try to find the | |
07:12 | volume of the triangular prism . Now again , the | |
07:15 | first step is to calculate the area of the base | |
07:18 | of the prism , which is the triangle . The | |
07:21 | formula for finding the area of a triangle is one | |
07:23 | half of the triangle's base times its height . Be | |
07:26 | careful not to confuse the base of the triangle with | |
07:29 | the base of the prism , which is the triangle | |
07:31 | itself . Our diagram tells us that the base of | |
07:34 | the triangle is 10 inches and its height is eight | |
07:36 | inches . So one half the base times the height | |
07:39 | would be one half of 10 times eight , 10 | |
07:42 | times eight is 80 and one half of 80 is | |
07:44 | 40 . So the area of the triangle is 40 | |
07:46 | inches squared . Now all we have to do to | |
07:49 | find the volume is to multiply that area by the | |
07:51 | length that it was extended . We're told that the | |
07:53 | length is 50 inches . So we multiply 40 inches | |
07:56 | squared by 50 inches and we get 2000 inches cubed | |
08:00 | . That's the volume of this prison . All right | |
08:03 | . Are you ready to try finding the volume of | |
08:05 | the cylinder ? Now the cylinder is made by extending | |
08:08 | a circle . So first we need to calculate the | |
08:10 | area of that circle . As we learned in a | |
08:12 | previous video . The area of a circle is found | |
08:15 | by multiplying pi times its radius squared . The radius | |
08:19 | of this circle is two m . So if we | |
08:21 | square that , we get four m squared , then | |
08:25 | we multiply by pi which is approximately 3.14 and we | |
08:28 | get 12.56 m squared . Now that we know the | |
08:32 | area of the circle to find the volume of the | |
08:34 | cylinder . We just multiply that area by the length | |
08:37 | that the circle was extended , Which happens to be | |
08:40 | 10 m 10 m , times 12.56 m squared , | |
08:45 | gives us 125.6 m cubed . So the volume of | |
08:49 | the cylinder is 125.6 m3 . So do you see | |
08:54 | how to find the volume of three D shapes ? | |
08:56 | Like these shapes ? That are made by taking a | |
08:58 | two D . Shape and then extending it along the | |
09:00 | third axis . You just find the area of that | |
09:02 | shape and then you multiply it by the length that | |
09:05 | it was extended . And that gives you the volume | |
09:07 | And it works for any two D shape . It | |
09:10 | works for trapezoid or pentagons or diamonds or stars or | |
09:14 | hearts . All right , man . Now that I | |
09:19 | know how to accurately calculate volumes . Playtime is way | |
09:22 | more fun . Of course not all three D shapes | |
09:26 | are made by extending a two D shape along a | |
09:28 | third dimension . Some are made by rotating the shape | |
09:31 | around an axis , like the sphere we saw earlier | |
09:34 | in the video , for instance , one way to | |
09:35 | form a sphere is to rotate a two D circle | |
09:38 | around one of its diameter lines and another common three | |
09:41 | D shape . That can be formed by rotating a | |
09:43 | two D shape around an axis is a cone . | |
09:46 | A cone can be formed by rotating a right triangle | |
09:49 | around one of its two perpendicular edges . Since the | |
09:52 | formulas for finding the volumes of spheres and cones are | |
09:55 | more complicated to explain . We don't have time to | |
09:58 | learn how we arrive at them in this video . | |
10:00 | So for now it's best if you just memorize the | |
10:02 | formulas so you can use them on tests if you | |
10:04 | need to to find the volume of a sphere . | |
10:06 | You use the formula volume equals four thirds times pi | |
10:11 | times radius cubed and find the volume of a cone | |
10:14 | . You use the formula volume equals one third times | |
10:17 | height times pi times radius squared . Let's try one | |
10:20 | example of each . Before we wrap up , Here's | |
10:23 | a sphere with a radius of 2cm . Fortunately that's | |
10:27 | the only dimension . We need to find its volume | |
10:29 | . Our formula says that the volume of a sphere | |
10:32 | is equal to four thirds times pi times radius cubed | |
10:36 | , cubing the radius means multiplying it by itself three | |
10:39 | times . So we take two centimeters times two centimeters | |
10:42 | times two centimeters , which equals eight centimeters cubed . | |
10:46 | Next we'll multiply that by our approximation for PIN 3.14 | |
10:51 | times eight is 25.12 And last we multiply that by | |
10:56 | 4/3 , which is the same as multiplying by four | |
10:59 | , then dividing by three , which gives us a | |
11:00 | volume of 33.49 cm cubed . Now let's try a | |
11:06 | cone to use our formula to find the volume of | |
11:08 | a cone . We need to know two things the | |
11:11 | radius of the circle that forms the base of the | |
11:13 | cone and the height of the cone , which is | |
11:15 | similar to the height of a triangle . It's the | |
11:17 | distance from the point at the tip of the cone | |
11:20 | , straight down to the center of the circular base | |
11:23 | . The radius of the base of this cone is | |
11:25 | three ft and its height is nine ft . So | |
11:28 | first let's plug that radius into our equation and square | |
11:31 | it three ft times three ft is nine ft squared | |
11:35 | . Next we multiply that by pi again using 3.14 | |
11:39 | and we get 28.26 ft squared . You may notice | |
11:43 | that that's just the area of the base circle , | |
11:46 | But now we need to multiply that base area by | |
11:49 | 1/3 times the height of the cone . One third | |
11:52 | times nine ft is three ft and three ft times | |
11:55 | 28.26 square feet gives us 84.78 cubic feet , which | |
12:00 | is the volume of the cone . All right now | |
12:03 | , you know a lot about volume . You know | |
12:05 | that it's a three dimensional quantity of geometric objects and | |
12:08 | you know that we measure volume with cubic units . | |
12:11 | You also learn how to calculate the volume of some | |
12:14 | basic three D shapes . Of course , there's a | |
12:16 | lot of other three D shapes that we didn't have | |
12:18 | time to cover in this video . But now , | |
12:20 | you know about some of the most common ones . | |
12:22 | The key is to put what you've learned into practice | |
12:25 | by trying some exercise problems on your own . As | |
12:28 | always , thanks for watching mathematics and I'll see you | |
12:30 | next time . Learn more at math Antics dot com |
Summarizer
DESCRIPTION:
OVERVIEW:
Math Antics - Volume is a free educational video by Mathantics.
This page not only allows students and teachers view Math Antics - Volume videos but also find engaging Sample Questions, Apps, Pins, Worksheets, Books related to the following topics.