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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