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  • - [Instructor] In this video, we're going

  • to talk about the volume of a pyramid.

  • And many of you might already be familiar

  • with the formula for the volume of a pyramid.

  • But the goal of this video is to give us an intuition

  • or to get us some arguments

  • as to why that is the formula for the volume of a pyramid.

  • So let's just start by drawing ourselves a pyramid.

  • And I'll draw one with a rectangular base.

  • But depending on how we look at the formula,

  • we could have a more general version.

  • But a pyramid looks something like this.

  • And you might get a sense of what the formula

  • for the volume of a pyramid might be.

  • If we say this dimension right over here is x.

  • This dimension right over here,

  • the length right over here is y.

  • And then you have a height of this pyramid.

  • If you were to go from the center straight to the top

  • or if you were to measure this distance right over here,

  • which is the height of the pyramid.

  • You'll just call that, let's call that z.

  • And so you might say well,

  • I'm dealing with three dimensions,

  • so maybe I'll multiply the three dimensions together

  • and that would give you volume in terms of units.

  • But if you just multiplied xy times z,

  • that would give volume of the entire rectangular prism

  • that contains the pyramid.

  • So that would give you the volume

  • of this thing, which is clearly bigger,

  • has a larger volume than the pyramid itself.

  • The pyramid is fully contained inside of it.

  • So this would be the tip of the pyramid on the surface,

  • it's just like that.

  • And so you might get a sense that, all right

  • maybe the volume of the pyramid is equal to x times y

  • times z, times some constant.

  • And what we're going to do in this video

  • is have an argument as to what that constant should be.

  • Assuming that this, the volume of the parameter

  • is roughly of the structure.

  • And to help us with that,

  • let's draw a larger rectangular prism

  • and break it up into six pyramids,

  • that completely make up the volume of the rectangular prism.

  • So first, let's imagine a pyramid that looks

  • something like this, where its width is x,

  • its depth is y, so that could be its base.

  • And its height is halfway up the rectangular prism.

  • So the rectangular prism has height z,

  • the pyramid's height is going to be z over two.

  • Now what would be the volume of the pyramid based

  • on what we just saw over here?

  • Well, that value would be equal to some constant k

  • times x, times y, not times z, times the height

  • of the pyramid, times z over two.

  • So it'd be x times y times z over two, I'll just write

  • times z over two or actually we can even write it this way

  • xy is z over two.

  • Now I can construct another pyramid

  • has the exact same dimensions.

  • If I were to just flip that existing pyramid on its head

  • and look something like this.

  • This pyramid also has dimensions of an x

  • width, a y depth and a z over two height.

  • So it's volume would be this as well.

  • Now what is the combined volume of these two pyramids?

  • Well, it's just going to be this times two.

  • So the combined volume of these pyramids,

  • let me just draw it that way.

  • So these two pyramids that look something like this,

  • I'm gonna try to color code it.

  • We have two of them.

  • So two times their volume,

  • is going to be equal to well two times this

  • is just going to be k times xyz.

  • Kxy and z.

  • And we have more pyramids to deal with for example,

  • I have this pyramid, right over here

  • where this face is its base

  • and then if I try to draw

  • pyramid it looks something like this,

  • this one right over there.

  • Now what is its volume going to be?

  • Its volume is going to be equal to k times its base is y

  • times z so kyz.

  • And what's its height?

  • Well, its height is going to be half of x.

  • So this height right over here is half of x.

  • So it's k times y times z times x over two

  • or I could say times x and then divide everything by two.

  • Now I have another pyramid

  • that has the exact same dimensions.

  • This one over here,

  • if I try to draw it on the other face,

  • opposite the one we just saw

  • essential if we just flip this one over,

  • has the exact same dimensions.

  • So one way to think about it,

  • we have two pyramids that look like that

  • with those types of dimensions.

  • This is for an arbitrary rectangular prism

  • that we are dealing with.

  • So I have two of these,

  • and so if you have two of their volumes,

  • what's it going to be?

  • It's just going to be two times this expression.

  • So it's going to be k times xyz.

  • xyz, interesting.

  • And then last but not least we have two more pyramids.

  • We have this one, that has a face, that has the base

  • right over here, that's its base

  • and if it was transparent you'd be able to see

  • where I'm drawing right here.

  • And then you have one on the opposite side,

  • right over, there on the other side.

  • Like as if you were to flip this around.

  • And so by the exact same argument,

  • so let me just draw it.

  • So we have two of these, two of these pyramids

  • my best to draw it so times two.

  • So each of them would have a volume of what?

  • Each of them their base is x times z.

  • So it's going to be k times x times z

  • that's the area of their base.

  • And then what is their height?

  • Well, each of them has a height of y over two.

  • So times y over two and I have two of those pyramids.

  • So I'm going to multiply those by two,

  • the twos cancel out so I'm just left with k times xyz.

  • So k times xyz.

  • Now one of the interesting things

  • that we've just stumbled on in this,

  • is seeing that even though these pyramids

  • have different dimensions and look different,

  • they all have actually the same volume

  • which is interesting in and of themselves.

  • And so if we were to add up the volumes

  • of all of the pyramids here and use this formula

  • to express them, so if I were to add all of them together

  • that should be equal to the volume

  • of the entire rectangular prism.

  • And then maybe we can figure out k.

  • So the volume of the entire rectangular prism is xyz.

  • X times y times z

  • and then that's got to be equal to the sum of these.

  • So that's going to be equal to kxyz plus kxyz

  • plus kxyz or you could say

  • that's going to be equal to three kxyz.

  • All I did is, let me just add up the volume

  • from all of these pyramids.

  • And so what do we get for k?

  • Well, we could divide both sides by three xyz

  • to solve for k, three xyz.

  • Three xyz and we are left with on the right hand side

  • the everything cancels out we're just left with a k.

  • And on the left hand side we're left with a 1/3.

  • And so we get k is equal to 1/3, K is equal to 1/3

  • and there you have it, that's our argument

  • for why the volume of a pyramid is 1/3 times

  • the dimensions of the base, times the height.

  • So you might see it written that way

  • or you might see it written as 1/3 times base

  • and so if x times y is the base, so the area of the base,

  • so the base area times the height

  • which in this case is z,

  • but if you say h for that,

  • you might see the formula for a pyramid

  • written this way as well.

  • But they are equivalent,

  • but that's why you should feel good about the 1/3 part.

- [Instructor] In this video, we're going

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B1 中級

金字塔體積背後的直覺 (Intuition behind volume of a pyramid)

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    林宜悉 發佈於 2021 年 01 月 14 日
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