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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

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