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  • Hi! Welcome to Math Antics.

  • We've learned a lot about Geometry so far,

  • but there's one really important geometric shape that we still need to cover,

  • and that shape is a circle.

  • Since the invention of the wheel,

  • circles have been extremely important to all humanity.

  • Grog make wheel.

  • Thanks Grog!

  • In fact, you probably see circles almost everywhere you turn

  • But mathematically, what is a circle?

  • Well in Geometry, a circle is defined as:

  • the set of all points that are equidistant (or the same distance) from another single point.

  • And the best way to understand what that means is to see it in action.

  • Sohere's a single point to start with.

  • Now let's start drawing points that are equidistant from it.

  • This point is a foot away to the right.

  • Now let's make another point a foot away but in another direction. Let's say up here.

  • Now let's make another one, also a foot away, but in another direction. Right here.

  • Now let's make another, right here

  • and another, and another, and another, and another...

  • Wheew… I'm getting tired!

  • But do you see what's happening?

  • The more equidistant points we add, the more the pattern looks like a circle.

  • That's why a circle is defined as the set of points that are equidistant from a center point.

  • But of course, we usually don't see it as a set of points because there are infinitely many of them,

  • so they form a continuous circle.

  • Okay, now let's learn about the parts that make up a circle.

  • First of all we have the original point that we started with.

  • That's called the center, or the origin of the circle.

  • Next, we have the distance that we used to draw all of the equidistant point that form the circle.

  • That distance is called the radius.

  • The radius is important because

  • it's the distance from the center of a circle to ANY other point on the perimeter of that circle.

  • And even though a circle only has one radius dimension, you can draw as many radius lines as you want to.

  • Usually you'll only see one radius line drawn since it's the same length no matter where you draw it.

  • Another important circle dimension is called the diameter.

  • The diameter is the distance across a circle.

  • If you start at one point on the circle and then draw a line straight through the center to the other side,

  • that distance is the diameter.

  • As you can see, the diameter is really just the same as two radius lines drawn in exactly opposite directions.

  • So, for any circle, the diameter is always exactly twice as long as the radius.

  • All of the equidistant points we drew combine to form the perimeter of the circle.

  • Remember that perimeter is just the distance all the way around a shape.

  • But because a circle is a special shape, the perimeter of a circle gets a special name.

  • It's called the circumference.

  • The circumference is the distance all the way around a circle.

  • We're going to learn how to calculate the circumference of any circle in the next video.

  • We'll also learn how to calculate the area of any circle.

  • But before we can learn those things, we first need to learn about Pi.

  • Grog make Pie!

  • Sorry Grog, not that kind of pie.

  • In math, the word Pi (which is spelled 'P' 'i') refers to a very special number.

  • In fact, it's so special that it gets its own symbol.

  • This greek letter here is the symbol for the number Pi.

  • But... if Pi is just a number, why don't we write it like that?

  • Why do we use a special symbol for it?

  • That's a good question.

  • And I'll get to that in just a minute.

  • But first, let's learn what Pi really is by seeing how it relates to a circle.

  • It turns out that Pi is a really a Ratio!

  • Now if you're not sure what a ratio is, you can watch our video about them.

  • But basically, a ratio is just a relationship between two numbers that is written like a fraction.

  • Pi is the ratio of two different distances on a circle.

  • It's the ratio of the distance around a circle to the distance across a circle.

  • And what do we call those two distances?

  • Yep, the circumference and the diameter.

  • So Pi is the relationship of the circumference to the diameter.

  • And as you'll see in a minute, because Pi is a ratio,

  • it's the same number for any circle, no matter how big or small.

  • Okay, but what number is it?

  • What's the value of Pi?

  • Well, to figure that out, have a look at these two circles,

  • one big and one small.

  • We're going to imagine that our circles' diameters are flexible, like a piece of string,

  • and that we can wrap them around the outside edges (circumferences) of the circles.

  • So for each circle, if we start at the top and wrap the diameter around the circumference,

  • we see that 1 diameter is not enough to go all the way around.

  • So, let's get another diameter and keep going where the first diameter stopped.

  • Hmmmtwo diameters still isn't enough to go all the way around.

  • It looks like we're going to need to get a third diameter and keep going.

  • Awwww! So close!!

  • Three diameters is almost enough but it looks like

  • we're going to need just a little bit more to form a full circumference.

  • That little bit more turns out to be about 0.14 diameters.

  • That means that it takes 3.14 diameters to equal one circumference for any circle, big or small.

  • So the value of Pi is always 3.14.

  • Well okayPi is a little more complicated than that.

  • 3.14 is really just Pi rounded off to two decimal places.

  • And we actually have to round Pi off because it's a type of number that's called 'irrational'.

  • An irrational number has decimal digits that never end and never repeat.

  • Grog confused.

  • Yes, 'irrational' numbers are confusing,

  • but seeing some more of Pi's decimal digits will help you understand what I mean.

  • To be more precise, Pi is 3.141592653589793238…

  • and the decimal digits keep on going forever without repeating!!

  • Pretty amazing, huh?

  • But the good news is that saying Pi is 3.14 is usually close enough for most math problems,

  • so that's all you really need to memorize.

  • And that's why we use a symbol for Pi in equations.

  • We could write Pi with just two decimal places.

  • Or we could write it with 5 decimal places to be more accurate.

  • Or, we could write it with hundreds of decimal places to be super accurate.

  • Or, we could just use the symbol to represent the true value, which is infinitely accurate.

  • Okay, so in this video, we've learned what a circle is,

  • and we've learned about the important parts of a circle:

  • the center, the radius, the diameter and the circumference.

  • We've also learned about a very special number called Pi.

  • Pi is the ratio of a circle's circumference to its diameter,

  • and its value is about 3.14 no matter what size the circle is.

  • In our next video about circles, we're going to learn how

  • we can use the number PI to find the circumference and the area of any circle.

  • And even though there is not much math you can actually practice in this section, don't worry

  • there will be lots of practice problems in the next section to make up for it!

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

  • Mmmm, Grog good at math!

  • Learn more at www.mathantics.com

Hi! Welcome to Math Antics.

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Math Antics - 圓,什麼是PI? (Math Antics - Circles, What Is PI?)

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    Yassion Liu 發佈於 2021 年 01 月 14 日
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