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### Course: Get ready for AP® Calculus > Unit 7

Lesson 7: 2D vs. 3D objects# Dilating in 3D

The cross sections of 3D shapes are dilations of the original shape, centered at a specific point. The scale factor of the dilation depends on the height of the cross-section or the distance from the point on the base. Created by Sal Khan.

## Want to join the conversation?

- Would the dilations and cross sections look the same if you put the 3D pyramid on a coordinate plane?(14 votes)
- How can you put a 3D object such as a pyramid on a coordinate plane which is 2 dimensional? You could put it within a 3D space on an x-y-z coordinate system, but not a coordinate plane.(9 votes)

- Would it look the same if it was a 3D pyramid?(5 votes)
- I'm sorry, but a pyramid is always 3D, so your question makes no sense.(5 votes)

- im falling asleep(3 votes)
- So dont!

You must pay attention if you're going to learn anything from Sal!(6 votes)

- why does it keep asking for more questions(3 votes)
- هل يمكن لأي شخص أن يشرح لي هذا في فترة أسبوعين(3 votes)
- Would the dilations and cross sections look the same if you put the 3D pyramid on a coordinate plane(1 vote)
- One, you copied @FemiO, two you can't put a 3D object on a coordinate plane.(3 votes)

- How do we know which dilation is a 0.5 or a 0.75?

i am so confused about this?(2 votes)- Think of the 0.5 dilation being the halfway point between points B and P going down. For the 0.75 dilation you use the same method, but you go further down (obviously). The closer you cut to the base of the pyramid, the bigger the dilation fraction is.(1 vote)

- Would the dilations and cross sections look the same if you put the 3D pyramid on a coordinate plane?(2 votes)
- How do you show a 3D figure on a 2D coordinate plane?(1 vote)

- so is it going to stay the same no matter what?(1 vote)
- Length wise yes, size whise no(1 vote)

- The pyramid cross-sections are really constrictions of the original triangle, not dilations--they are smaller than the original base triangle, not bigger. Using a word contrary to its meaning is confusing. Why do they do that?(1 vote)

## Video transcript

- [Instructor] Let's say I
have some type of a surface, let's say that this right over
here is the top of your desk. And I were to draw a
triangle on that surface. So maybe the triangle looks like this, something like this, it doesn't have to be a right triangle. And so I'm not implying that this is necessarily a right triangle. Although it looks a little bit like one, and let's call it
triangle A, B, and then C. Now what I'm going to do
is something interesting. I'm gonna take a fourth point P that's not on the surface of this desk and it's going to be right above point B. So let me just take that point, go straight up, and I'm going to get to
point P right over here. Now, what I can do is construct a pyramid using point P as the peak of that pyramid. Now, what we're going
to start thinking about is what happens if I take
cross sections of this pyramid? So in this case, the length of segment PB is
the height of this pyramid. Now, if we were to go
halfway along that height, and if we were to take a
cross section of this pyramid that is parallel to the
surface of our original desk, what would that look like? Well, it would look something like this. Now you might be noticing
something really interesting. If you were to translate that
blue triangle straight down onto the surface of the table, it would look like this. And when you see it that way, it looks like it is a dilation
of our original triangle centered at point B. And in fact, it is a
dilation centered at point B with a scale factor of 0.5. And you can see it right over here, this length right over here, what BC was dilated down
to is half the length of the original BC. This is half the length
of the original AB, and then this is half the
length of the original AC. But you could do it at other
heights along this pyramid. What if we were to go 0.75
of the way between P and B. So if you were to go right over here. So it's closer to our original triangle, closer to our surface. So then the cross section would look like this. Now, if we were to translate that down onto our original surface, what would that look like? Well, it would look like this. It would look like a dilation of our original triangle
centered at point B. But this time with a scale factor of 0.75. And then what if you were to
go only a quarter of the way between point P and point B? Well, then you would
see something like this, a quarter of the way. If you were take the cross section parallel to our original surface, it would look like this. If you were to translate that
straight down onto our table, it would look something like this. And it looks like a dilation centered at point B with
a scale factor of 0.25. And the reason why all of these dilations look like dilations centered at point B is because point P is
directly above point B. But this is a way to
conceptualize dilations, or see the relationship
between cross sections of a three-dimensional shape, in this case like a pyramid, and how those cross sections relate to the base of the pyramid. Now, let me ask you an
interesting question. What if I were to try
to take a cross section right at point P. Well, then I would just get a point. I would not get an actual triangle, but you could view that as a dilation with a scale factor of zero. And what if I were to take
a cross section at the base? Well, then that would be my
original triangle, triangle ABC. And then you can view that as a dilation with a scale factor of one 'cause you've gone all
the way down to the base. So hopefully this
connects some dots for you between cross sections of
a three-dimensional shape that is parallel to the base
and notions of dilation.