Placeholder Image

字幕列表 影片播放

  • Hi! Welcome to Math Antics.

  • In our last video, we learned about circles and we learned about a special ratio called Pi.

  • In this video, were going to learn how we can use that ratio to calculate the circumference and the area of any circle.

  • The formulas that we use to calculate circumference and area are so important that you should really memorize them.

  • To help you do that, we're going to look at them side-by-side.

  • That will help you see their similarities and their differences so you don't get them mixed up.

  • The formula for finding the circumference is:

  • Circumference equals Pi times diameter.

  • And just like most formulas, we use abbreviation:

  • 'C' for circumference and 'd' for diameter.

  • That's a pretty simple formula.

  • It tells us that if we know the diameter of a circle, all we have to do is

  • multiply that diameter times the number Pi and we'll get the circumference.

  • We'll try that formula out in a few minutes.

  • But first, let's see the formula for area.

  • The formula for finding the area of a circle is:

  • Area equals Pi times radius squared.

  • Again, we can use abbreviations to make it shorter:

  • 'A' for area and 'r' for radius.

  • Now this is a pretty simple formula too.

  • It tells us that if we know the radius,

  • we just have to 'square' it and then multiply that times Pi to get the area.

  • Okay, but what does it mean to 'square' the radius?

  • Well, squaring a number just means multiplying it by itself.

  • For example, 3 squared just means 3 times 3,

  • and 5 squared just means 5 times 5

  • and r squared just means r times r.

  • So our formula is really just:

  • Area equals Pi times r times r, but we write it in the 'r squared' form because it's more compact.

  • Oh, and one really important thing to keep in mind is that

  • r squared is NOT the same thing as 2 times r.

  • That's a common mistake that students make when first learning how to find the area of a circle.

  • And if we look carefully at both of our formulas, you'll see why.

  • These two formulas have a lot in common.

  • In each of them, you are multiplying Pi by part of a circle to find either the circumference or the area.

  • In the case of the circumference, you are multiplying Pi times the diameter,

  • and in the case of area, you are multiplying Pi times the radius squared.

  • But do you remember the relationship between the radius and the diameter?

  • Diameter is just 2 times the radius.

  • So we could re-write our formula for circumference like this:

  • Circumference = Pi × 2 × r.

  • Ha! Now you see why it's so easy to get confused.

  • To find the circumference, you take the radius and double it.

  • Then you multiply by Pi to get the final answer.

  • But for area, you don't double the radiusyou square it.

  • That's a very important difference.

  • To help you see that difference in action,

  • let's find both the circumference and the area of this circle using our two formulas.

  • The only thing we know about this circle is that the radius is 8 meters.

  • Luckily, that's all we need to know.

  • First, we use our formula for circumference: C = Pi × d.

  • To get the diameter, we take the radius and we double it. …that is, we multiply it by 2.

  • 2 × 8 = 16, so the diameter is 16 meters.

  • Then, we multiply that by Pi to get the circumference.

  • Since this is decimal multiplication, I'm going to use a calculator.

  • 16 × 3.14 = 50.24

  • So that means that the circumference of this circle is 50.24 meters.

  • Alright, now let's find the area using our formula: A = Pi times r squared.

  • Again, we start with the radius, but instead of doubling it, we 'square' it.

  • That means we multiply it by itself.

  • 8 m × 8 m = 64 meters squared.

  • Then we multiply that by Pi.

  • 64 × 3.14 = 200.96 meters squared.

  • That's the area of this circle.

  • As you can see, the result we get when we square the radius

  • is very different from the result we get when we we double it.

  • And one of the most important differences is with the units of our answer.

  • Doubling the radius just gives us the diameter, which is a 1-dimensional quantity.

  • So, the answer we get from our formula for circumference is also a 1-dimensional quantity.

  • But, when we square the radius, that gives us 'square units', which are 2-dimensional.

  • That makes sense because area is always a 2-dimensional quantity.

  • Remembering that will help you avoid getting these two formulas mixed up.

  • The one that has the radius squared is always for area.

  • Alright, let's try a couple real-world examples to make sure you've got it.

  • Here's the real world, which as you probably know is a sphere.

  • But, if we take a slice of the world, right at the equator, that slice is a circle.

  • Let's find the circumference of that circle.

  • To do that, we need to know the diameter of the earth.

  • That turns out to be about 12,750 km.

  • Great, then to find the circumference we just need to multiply that diameter times Pi.

  • Now I'm definitely going to use a calculator for this.

  • And, I'm going to use a more accurate version of Pi since this is such a big distance.

  • 12,750 × 3.14159 = 40,055 km (to the nearest kilometer).

  • Wow, that's a pretty big circumference!

  • No wonder it takes so long to go all the way around the earth!

  • On your markGet setGo!

  • Whooo - Yes! 3.14 seconds quicker than last time. Yes!

  • Here's another real-world example with a circle.

  • If this pizza has a diameter of 24 inches, what's its total area?

  • Well, using our formula, we start by squaring the radius.

  • But, the problem didn’t give us the radius

  • it gave us the diameter, so we have to calculate the radius from the diameter.

  • Fortunately, that's really easy.

  • The radius is just half of the diameter, so we just need to divide the diameter by 2.

  • 24 inches divided by 2 gives us 12 inches for the radius.

  • And now that we know the radius, we need to square it.

  • 12 in × 12 in = 144 inches squared.

  • Next, we just multiply that by Pi.

  • 144 × 3.14 is 452.16.

  • So, the total area of the pizza is 452.16 square inches.

  • Alright, so know you know how to find the circumference and the area of any circle.

  • All you need to do is remember the formulas:

  • Circumference equals Pi times diameter,

  • and Area equals Pi times radius squared.

  • But, it's really important to practice using these formulas for yourself,

  • so be sure to try some of the exercises problems.

  • That's the way to really learn math.

  • Thanks for watching Math Antics and I'll see ya next time.

  • Learn more at www.mathantics.com

Hi! Welcome to Math Antics.

字幕與單字

影片操作 你可以在這邊進行「影片」的調整,以及「字幕」的顯示

B2 中高級 美國腔

Math Antics - 圓、圓周率和麵積 (Math Antics - Circles, Circumference And Area)

  • 17 7
    Yassion Liu 發佈於 2021 年 01 月 14 日
影片單字