Math Antics - Volume - Free Educational videos for Students in K-12 | Lumos Learning

Math Antics - Volume - Free Educational videos for Students in k-12


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
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