Math Antics - Circles, Circumference And Area - Free Educational videos for Students in K-12 | Lumos Learning

Math Antics - Circles, Circumference And Area - Free Educational videos for Students in k-12


Math Antics - Circles, Circumference And Area - By Mathantics



Transcript
00:03 Uh huh . Hi , welcome to Math Antics .
00:08 In our last video we learned about circles and we
00:11 learned about a special ratio called Pie . In this
00:14 video , we're going to learn how we can use
00:16 that ratio to calculate the circumference and the area of
00:19 any circle . The formulas that we use to calculate
00:23 circumference in area are so important that you should really
00:26 memorize them to help you do that . We're going
00:29 to look at them side by side and that will
00:31 help you see their similarities and their differences . So
00:34 you don't get them mixed up . The formula for
00:37 finding the circumference is circumference equals pi times diameter and
00:42 just like most formulas , we use abbreviations , see
00:46 for circumference and D for diameter . So that's a
00:50 pretty simple formula . It tells us that if we
00:52 know the diameter of a circle , all we have
00:54 to do is multiply that diameter times the number pi
00:58 and we'll get the circumference . Now we'll try that
01:00 formula out in a few minutes . But first let's
01:03 see the formula for area . The formula for finding
01:07 the area of a circle is area equals pi times
01:11 radius squared . Again , we can use abbreviations to
01:15 make it shorter . A for area and are for
01:17 radius . Now this is a pretty simple formula two
01:21 . It tells us that if we know the radius
01:23 , we just have to square it and then multiply
01:26 that times pi to get the area . Okay ?
01:30 But what does it mean to square the radius ?
01:33 Well , squaring the number just means multiplying it by
01:36 itself . For example , three squared just means three
01:41 times three and five squared just means five times 5
01:46 and R squared just means our times are . So
01:50 our formula is really just area equals pi . Times
01:54 are times are , but we write it in the
01:57 R squared form because it's more compact . Oh and
02:02 one really important thing to keep in mind is that
02:05 R squared is not the same thing as two times
02:09 are . That's a common mistake that students make when
02:13 first learning how to find the area of a circle
02:15 . And if we look carefully at both of our
02:17 formulas , you'll see why these two formulas have a
02:21 lot in common in each of them . You're multiplying
02:24 pie by part of a circle to find either the
02:27 circumference or the area . In the case of the
02:30 circumference , you're multiplying pi times the diameter . And
02:33 in the case of the area you're multiplying pi times
02:37 the radius squared . But do you remember the relationship
02:41 between the radius and the diameter diameter is just two
02:44 times the radius . So we could rewrite our formula
02:48 for circumference . Like this , circumference equals pi times
02:52 two times are . Now you see why it's so
02:57 easy to get confused to find the circumference . You
03:00 take the radius and double it . Then you multiply
03:03 by pi to get the final answer . But for
03:06 area you don't double the radius , you square it
03:09 . And that's a very important difference . To help
03:13 you see that difference in action , lets find both
03:15 the circumference and the area of this circle . Using
03:19 our two formulas . The only thing we know about
03:21 the circle is that the radius is eight m ,
03:24 luckily that's all we need to know . First we
03:28 use our formula for circumference , circumference equals pi times
03:32 diameter to get the diameter we take the radius and
03:36 we double it . That is , we multiply it
03:38 by two , Two times eight equals 16 . So
03:41 the diameter is 16 m . Then we multiply that
03:46 by pi to get the circumference . Since this is
03:49 decimal multiplication , I'm going to use a calculator 16
03:53 times 3.14 equals 50.24 . So that means that the
03:58 circumference of the circle is 50.24 m . All right
04:03 now let's find the area . Using our formula area
04:07 equals pi times r squared . Again , we start
04:11 with the radius but instead of doubling it , we
04:13 square it . That means we multiply it by itself
04:17 eight m times eight m equals 64 m squared .
04:22 Then we multiply that by pi 64 times 3.14 equals
04:28 200.96 m squared . That's the area of this circle
04:34 . As you can see the result we get when
04:37 we square the radius is very different from the result
04:40 we get when we double it and one of the
04:42 most important differences is with the units of our answer
04:46 doubling the radius just gives us the diameter , which
04:49 is a one dimensional quantity . So the answer we
04:52 get from our formula for circumference is also a one
04:55 dimensional quantity , but when we square the radius ,
04:59 that gives us square units , which are two dimensional
05:02 , that makes sense because area is always a two
05:05 dimensional quantity . Remembering that will help you avoid getting
05:09 these two formulas mixed up . The one that has
05:12 the radius squared is always for area . Alright ,
05:17 let's try a couple real world examples to make sure
05:19 you've got it . Here is the real world ,
05:21 which , as you probably know , is a sphere
05:24 . But if we take a slice of the world
05:26 right at the equator that slices a circle , let's
05:29 find the circumference of that circle . To do that
05:33 . We need to know the diameter of the earth
05:35 . That turns out to be about 12,750 km .
05:40 Great . Then define the circumference . We just need
05:43 to multiply that diameter times . Pi . Now I'm
05:46 definitely going to use a calculator for this and I'm
05:49 going to use a more accurate version of pie since
05:52 this is such a big distance . So 12,750 times
05:58 3.14159 equals 40,000 and 55 kilometers to the nearest kilometer
06:06 . Wow , that's a pretty big circumference . No
06:10 wonder it takes so long to go all the way
06:12 around the earth . Your mark . It's it .
06:15 Go Who ? Yes , 3.14 seconds quicker than last
06:24 time . Yes . Who ? Here's another real world
06:28 example with a circle . If this pizza has a
06:31 diameter of 24 inches , what's its total area ?
06:35 Well , using our formula , we start by squaring
06:38 the radius , but the problem didn't give us a
06:41 radius that gave us the diameter , so we have
06:44 to calculate the radius from the diameter , fortunately ,
06:47 that's really easy . The radius is just half of
06:50 the diameter . So we just need to divide the
06:53 diameter by two 24 divided by two , gives us
06:57 12 for the radius . And now that we know
07:00 the radius , we need to square it 12 times
07:03 12 equals 144 squared . Next we just multiply that
07:09 by pi 144 times 3.14 is 452.16 So the total
07:18 area of the pizza is 452.16 square inches . All
07:24 right , so now you know how to find the
07:26 circumference and the area of any circle . All you
07:30 need to do is remember the formulas circumference equals pi
07:34 times diameter and area equals pi times radius squared .
07:39 But it's really important to practice using these formulas for
07:42 yourself . So be sure to try some of the
07:44 exercise problems . That's the way to really learn math
07:48 . Thanks for watching Math Antics and I'll see you
07:50 next time learn more at Math Antics dot com .
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