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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.

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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.