Volume through decomposition

- [Voiceover] Let's see if we can figure out the volume of this figure over here. They've given us some of the dimensions. We see this side over here is two centimeters, this is seven centimeters, this is 12 centimeters, this is five centimeters, this is three centimeters. And so like always, pause this video and see if you can figure it out. Well there's a bunch of ways to do this, but the way I'd like to do it is just to break it up into two rectangular prisms. So what I'm gonna do is, in fact most of the reasonable ways to do this would be to break it up into two rectangular prisms, and the ones that jump out at me is one prism like this that is three centimeters wide, five centimeters high, and then it is seven centimeters long, or seven centimeters deep. So this one right over here. And if this part right over here was transparent you would see it look just like this. You would see it look just like this. And so this one once again, it is three centimeters wide, seven centimeters long. So this distance right over here is going to be the same as this distance right over here. So seven centimeters long. So the width times the length times the height is five centimeters. Gets us to, let's see. Three times seven is 21, times five is equal to, 20 times five is 100, one times five is five. So it's going to be 105. We can say 105 cubic centimeters, cause you have centimeters times centimeters times centimeters. So this blue part right over here, this blue rectangular prism, has a volume of 105 cubic centimeters. So now we can separately figure out the volume of what I'm now highlighting in this magenta color. What I'm highlighting in this magenta color. If this was transparent, you would see this part back over here and right over here. So what are its dimensions? Well, we know its height is two centimeters, we know that this dimension right over here, I guess you could say its depth, we could call it that, is seven centimeters. But what is this right over here? If we want to consider this, maybe it's length, or maybe it's width, depending on what we want to call it. Well, let's see, this whole thing is 12 centimeters, from here to here is 12 centimeters, and we know that from here to here is three centimeters, so this piece right over here must be nine centimeters. So that must be nine centimeters, is this distance right over here. So the volume of this magenta part is going to be nine centimeters times seven centimers times the height, times two centimeters. Which is going to get us, let's see, nine times seven is 63, 63 times two is equal to, 60 times two is 120, three times two is six, so it's 126 cubic centimeters. So the total volume of the entire thing is going to be the volume of the magenta stuff, which is 126 cubic centimeters, plus the volume of the blue stuff, plus 105 cubic centimeters. And that's going to give us, for the entire figure, six plus five is 11, so one plus two is three, that's really one ten plus two tens is three tens. And then 100 plus 100 is 200, so we get 231 cubic centimeters is the volume of the entire thing. Fascinating.