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### Course: 6th grade (Illustrative Mathematics) > Unit 1

Lesson 11: Extra practice: Surface area# Surface area word problem example

Akira won a science fair and got a golden pyramid trophy. We want to find out how much gold foil was used. The trophy has a square base and four triangular faces. By calculating the area of these shapes, we can find the total surface area covered by the gold foil.

## Want to join the conversation?

- why do you have to multiply by 1/2?😕(13 votes)
- Since the triangle is half of a parallelogram, which is why you multiply it by 1/2.(29 votes)

- Any time you have a 3d surface area, do you have to unfold it all the time to find the area?(10 votes)
- No you can use formulas.(14 votes)

- I don’t get what lateral area is? Please help.(6 votes)
- It’s basically the surface area but without the base/s included(11 votes)

- Why do i have to do 5x3 if im finding the area of a box that has a height of 5 and a width of 3?(7 votes)
- b/c you find area by multiplying length and height(5 votes)

- Why do you have to combine the area of both figures?(5 votes)
- Because you are trying to find the total area.(3 votes)

- this is very good. my grandma is a khan academy ambassador(5 votes)
- how do you find the height of a pyramid if you are given the slant height?(5 votes)
- The blue dotted line with the little square indicates it is the true height. You would not be able to have a slant with a 90 degree angle. The solid lines are the slants for the figure(1 vote)

- this is hard i had this for my homework and still i dont get it(4 votes)
- It's hard I agree. Try watching this and it might help you. https://youtu.be/7uzSWG6sRH0. Copy this and paste it to your search bar. It helped me better.(2 votes)

- I don’t understand surface area. Can someone please help me just so I know more about surface area?🤔(2 votes)
- Surface area is the area of the surface. In the video, Sal makes a net, basically making it flat, then calculating the area of each part and adding them up(2 votes)

- and I am dumb so I understand none of this

;-;(1 vote)- You are not dumb; I used to not understand it either. Don't worry about your problems, just keep trying or search it up!(7 votes)

## Video transcript

- [Instructor] Akira receives
a prize at a science fair for having the most informative project. Her trophy is in the
shape of a square pyramid and is covered in shiny gold foil. So this is what her trophy looks like. How much gold foil did it
take to cover the trophy, including the bottom? And so they give us some dimensions and we want how much gold foil, and it's an inches squared, so it's really going to be an area. So pause this video and see
if you can figure that out. How much gold foil did it
take to cover the trophy? All right, now let's work
through this together. And so essentially, what they're asking is what is the surface area
of this square pyramid? And we're gonna include the base, 'cause that surface area is how much, it's the area of the
gold foil that is needed. Now, sometimes, some of you might be able to think about this just
by looking at this figure, but just to make sure
we don't miss any area, I'm gonna open up this square pyramid and think about it in two dimensions. So what we're gonna do is
imagine if I were to unleash, or if I were to cut the top
and, let me do this in red, if I were to cut this edge,
if I were to cut this edge, if I were to cut that edge, and that edge, so the edges that connect the triangular sides, and if
I were to just open it all up, what would this look like? So if I were to open it all up. Well at the bottom, you
would have your square base. Let me color that in. So
you have your square base. So let me draw that. So you have your square base. This is gonna be a rough drawing. And what are the dimensions
there? It's three by three. We know this is a square pyramid, so the base, all the
sides are the same length. They give us one side, but
then if this is three inches, and this is gonna be three inches as well. And let me color that same
color, just so we recognize that we're talking about this same base. And if we open up the triangular faces, what's it going to look like? Well, this is going to look like this. This is a rough hand drawing,
but hopefully it makes sense. This is going to look like this. And each of these triangular faces, they all have the exact same area, and the reason why I know that, they all have the same base, three, and they all have the
same height, six inches, but I'll draw that in a second. So they all look something like
this, just hand drawing it. And all of their heights, all of their heights are six inches. So this right over here is six inches. This over here is six inches. This over here is six inches. And this over here is six inches. So to figure out how
much gold foil we need, we're trying to figure
out the surface area, which is really just
gonna be the combined area of these figures. Well, the area of this central square is pretty easy to figure out. It's three inches by three inches, so it would be nine inches,
nine inches squared. Now, what are the area of the triangles? Well, we could figure out the
area of one of the triangles and then multiply by four
since there are four triangles. So the area of this
triangle right over here, it's gonna be one half times our base, which is three times three
times our height, which is six. Let's see, one half times three times six. That's one half times 18,
which is equal to nine, nine square inches or nine inches squared. So what's gonna be our total area? Well, you have the area
of your square base plus you have the four sides, which each have an area of
nine, so I could write it out. I could write four times
nine, or I could write nine. Do that black color, or I could write nine plus nine plus nine plus nine. And just to remind ourselves, this is, that right over there, is the area of one triangular face. Triangular face. So this is all of the triangular faces. Triangular faces, and of
course ,we have to add that to the area of their square base. So this is nine plus nine times four. You could view this as nine times five, which is gonna be 45 square inches, nine plus nine plus nine plus nine.