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Surface area of a box using nets

Video transcript
- [Voiceover] In a previous video, we figured out how to find the surface area of this cereal box by figuring out the areas of each of the six surfaces of the box, and then adding them all up. I'm going to do that again in this video, but I'm going to do it by visualizing a net for the box. The way I think about a net of a box like this, is what would happen if you were to cut the cardboard and then flatten it all out? So what am I talking about? Well, what we have here, we could imagine making a cut in the box, and the cut could be, let's see, I could make a cut back here, So I could make a cut right over there, I could cut it... I could cut it right like this, I could cut it like that. So if I just did that, this top flap would flap open so that would be able to come out like that, and then I could also make a cut for this side. So I could make a cut back there, and I could make a cut right over here, and now this side could flap forward, and I could do the same thing on this other side right over here. Then that could flap forward, and then, the back side, I could draw it. So I would also have a cut. I'll draw it as a dotted line because you're not supposed to be able to see this cut. But the corresponding cut to this one on the side that we can't see, we can draw it a little bit neater than that. The corresponding cut would be right back here. Right back there, and then a cut right over here. And so what would happen if we were to flatten all of this out? Well we would have, we would have the front of the box. I'll try to draw this as neatly as I can. So the front of the box looks like this. We would have this top flap, which looks like this If we were to flatten it all out. We have these two side flaps. So that's a side flap. That's a slide flap. And this is another, this side flap right over here. That's a side flap. Then we would have the bottom of the box. So the bottom of the box is going to look like this. The bottom of the box. And then we have the back of the box. That the bottom is going to be connected to. We didn't cut that. So we have the back of the box. The back of the box looks like this. And there we have it. We've made the net. This is what would happen if you made the cuts that I talked about and then flatten the box out, it would look like this. Now how can we use this net to find the surface area? Well we just need to figure out the surface area of this shape now. So how do we do that? Well we know a lot about the dimensions. We know that this width right over here, that this is ten centimeters. Ten centimeters from there to there. We know the height, actually, going all the way from here all the way up. Because the height of the box is 20 centimeters, so this is going to be 20 centimeters right over here. And you have another 20 centimeters, you have another 20 centimeters right over here. And right over here if you like. And then you have, see the depth of the box is three centimeters, so this is three centimeters. Three centimeters, and then this is three centimeters. And so what is the area, actually just let me do one region first. What is the area of this entire region that I am shading in with this blue color? Well it's ten centimeters, that is, I'll do a color that you can see a bit more easily. It is ten centimeters wide, ten centimeters. Times, what's the height? 20, plus three, plus 20, plus three. So that's going to be 40 plus six, so times 46 centimeters. That's this blue area. So that's going to be 460, 460 square centimeters. 460 square centimeters, and now we just have to figure out the area of the two flaps. So this flap right over here is 20 centimeters by three centimeters. So that's 60 centimeters squared. So 60 centimeters squared, or 60 square centimeters I should say. And then this flap is going to have the exact same area. Another 60 square centimeters. 60 square centimeters, and you add everything together. We deserve a little bit of a drumroll. We get, well this is going to add up to 580 square centimeters. Which is the exact thing we got in the other video where we didn't use a net, and you should just, it's nice to be able to do it either way. To be able to visualize the net or to be able to look at this and think about the different sides, even the sides that you might not necessarily see.