Surface area word problem example

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?

So they give us some dimensions, and we want how much gold foil, and it's in square inches. So, it's really going to be an area. 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. Essentially, what they're asking is: what is the surface area of this square pyramid? We’re going to include the base because that surface area is how much 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 going to open up this square pyramid and think about it in two dimensions.

What we're going to do is imagine if I were to unleash, or if I were to cut the top... 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—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. You have your square base; this is going to be a rough drawing.

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 3 inches, then this is going to be 3 inches as well. Let me color it that same color just so we recognize that we're talking about this same base.

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.

The reason why I know that they all have the same base is: 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 6 inches. This over here is 6 inches. This over here is 6 inches, and this over here is 6 inches. So to figure out how much gold foil we need, we're trying to figure out the surface area, which is really just going to be the combined area of these figures.

Well, the area of this central square is pretty easy to figure out: it's 3 inches by 3 inches, so it would be 9 square inches. Now, what are the areas 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 is going to be 2 times our base, which is 3 times 3 times our height, which is 6. Let's see: 1, 12 * 3 * 6—that's 1, 12 * 18, which is equal to 9 square inches or 9 inches squared.

So what's going to 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 9, or I could write 9—do that in black color—or I could write 9 + 9 + 9 + 9, and just to remind ourselves, this is that right over there is the area of one triangular face: triangular, triangular face.

So this is all of the triangular faces, and of course, we have to add that to the area of our square base. So this is 9 plus 9 times 4, or you could view this as 9 times 5, which is going to be 45 square inches: 9 + 9 + 9 + 9 + 9.