Main content

## High school geometry

### Course: High school geometry > Unit 9

Lesson 2: Cavalieri's principle and dissection methods- Cavalieri's principle in 2D
- Cavalieri's principle in 3D
- Cavalieri's principle in 3D
- Apply Cavalieri's principle
- Volume of pyramids intuition
- Volume of a pyramid or cone
- Volumes of cones intuition
- Using related volumes
- Use related volumes
- Volume of prisms and pyramids

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Using related volumes

CCSS.Math:

Use related volumes (using the volume of one figure to determine the volume of another). Created by Sal Khan.

## Want to join the conversation?

- I have a question about the last figure E. What is the meaning of
**by a scale factor of 1.5**? I thought it should be 1.5 times the base area, not the side length.(9 votes)- I got that wrong too.
**Dilation/Scale Factor**is by scaling its coordinates or lengths.(2 votes)

- If you are struggling with “Use related volumes” practice, like I was a few hours ago 😂…

Maybe this example could be useful for you:*We have:*

Square Prism A, its base is a square and each side of the square = 2cm

Square Prism B, its base is a square and each side of the square = 4cm

Both have same height = 6cm

Square Prism A volume = 24cm^3*We want to know:*

Square Prism B volume*Solution:*

You might be tempted to try this: ok 4/2 = 2, then scale factor = 2, then 24cm^3*2 = good answer. This is wrong because you used the sides’ scale factor (One-dimensional) to scale the whole shape (Three-dimensional). 🤯

This is how I would solve it:

We have a square at the bottom of the prism A, each side = 2cm

The area of this square will be 2cm * 2cm = 4cm^2

Also, we have a square at the bottom of the prism B, each side = 4cm

The area of this square will be 4cm * 4cm = 16cm^2**So, scaling by 2 our sides means that we are scaling by 2^2 our area**!

Now that we have had scaled properly our area, we can multiply it by our height (6cm) to know the total volume:

16cm^2 * 6cm = 96cm^3

Hopefully, this is helpful to someone! 😄(5 votes) - You should put this in the Tips and Thanks section and tell Sal what you just wrote!

It will be very helpful!

Have a BLESSED day!

M.L.M.(1 vote)

- When is 1.5x² equal to 1.5²x²?(1 vote)
- Those are never equal. The exponent is only on the "x" in the verstion on the left. If you start with (1.5x)², then it is equal to 1.5²x² because the exponent applies to both values.

Hope this helps.(2 votes)

- Can someone please explain how Sal solved for the volume of Figure E?

I am very confused!

Thanks in advance and have a BLESSED day!(1 vote) - The easiest numbers to calculate would be x=2 and h=7. And the last figure would have a volume of 63 fyi.(1 vote)
- the goofy goober(0 votes)
- hello my geeks and peeps my explainers and entertainers Rebecca Pahram here(0 votes)

## Video transcript

- [Instructor] We're told that all of the following figures
have the same height. All of the figures except
for B have square bases. So that's a square base, that's a square, that's a square, and that's a square. All of the figures
except for C are prisms. Yeah, C is a pyramid right over here. All of the figures except for D are right. You can see D right over
here is a little bit skewed, or you can view it as oblique. All of the figures except for
E have the same base area. The base of figure E is a
dilation of the base of figure A by a scale factor of 1.5. All right, so it tells a figure A has a volume of 28 cubic centimeters. What are the volumes of the other figures? So I'll pause this video and see if you can have a go with that. All right, now let's work
through this together. Now they're telling us about
the bases and the heights that a lot of these
have the same base area, figure E's going to be different. And they also tell us they
all have the same height. So one way to think about volume is it's going to deal
with base and height. And so for figure A, it's pretty straight forward. If we call this area right over here, let's call that b, for the area of the base, and then it has some
height, h, right over here. We know that the base
area times the height is going to be the volume. So we can say that based on figure A, base times height is going to be equal to 28 cubic centimeters. Fair enough. Now what's going on
over here with figure B? Well, it's a cylinder. Now for a cylinder, what is
the volume of a cylinder? Well, it, too, is going
to be base times height. So it's going to be the area
of the base times the height. And if you're wondering, "How is that possible that it's the same as a volume of a rectangular
prism over here?" It's actually Cavalieri's principle. If they have the same height and if at any point on that height, they have the same cross-sectional area, then you're going to have the same volume. So this volume is also going
to be base times height. So let me just say this is figure A, figure B right over here, let me draw those dots a little better, these colons a little bit better. Figure B, the volume is also
going to be base times height, which is equal to 28 cubic centimeters. Let me make that clear. That's the volume's equals to that. Volume is equal to that. Now what about for figure C? What is the volume going to be, what is the formula for
the volume of a pyramid? And we've gotten the intuition and proven this to
ourselves in other videos. Well, we know that for pyramid, the volume is going to be equal to 1/3 times base times height. And we know that it has the same base area as these other characters here, it has the same height. And so we know what base times height is. It's 28 cubic centimeters. So this is going to be 1/3
times 28 cubic centimeters. So this is going to be equal to 1/3 times 28 cubic centimeters, which we could rewrite as 28
over three cubic centimeters. You could also write that as
nine and 1/3 cubic centimeters. So that's for figure C. Now let's think about figure D. I'll do that right over here. Well, for this oblique prism, I guess we could say, you're going to have the same idea that comes from Cavalieri's
principle again. It's going to have the same formula for volume as figure A. It's going to be the base times
the height right over here. So I could write volume is
going to be equal to base times height and we
already know what that is. They tell us. The base times the height is going to be the same as figure A. It's going to be 28 cubic centimeters. Now let's go to figure E. This is an interesting one 'cause it has a different base area. What is gonna be the area right over here? Well, they tell us that the base of figure E is a dilation of the base of figure A
by a scale factor of 1.5. And these are both squares. So figure A, we'll say, x by x. This one over here is
going to be 1.5x by 1.5x. So let me write that down. 1.5x times 1.5x. Or another way to think about it, let me do it over here where I have some free space. We know that b, which we
know is an area of figure A, that would be equal to x times x. Now what's the area of
the base of figure E? Well, it's going to be 1.5x times 1.5x or 1.5x squared, which is the same thing as 1.5 squared is 2.25x squared, and we know x squared or x times x that is equal to b. That is equal to our original base area in all of these other figures. So the area over here, this area right over here is going to be 2.25 times b. 2.25 times b. Now that wasn't so easy to read. Let me write that a little bit clearer. So 2.25b is the base area right over here. And so what's the volume of this figure? The volume is going to
be the area of the base, which is 2.25 times the area of all these other figures' bases times the height, which
is the same, times h. Now we know what base times, we know what b times h is. Where b is the area of figure, the base area of figure A. We know that b times h
is 28 cubic centimeters. So the volume for figure
E is going to be 2.25 times 28 cubic centimeters, times 28 cubic centimeters. And I don't have a calculator
here in front of me and I can do it by hand, but I think you get the general point. You just have to multiply 2.25 times 28 to get the cubic, you get the volume of figure E. And that's because its
base has been scaled in each dimension by 1.5.