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Area of composite shapes

We can sometimes calculate the area of a complex shape by dividing it into smaller, more manageable parts. In this example, we can determine the area of two triangles, a rectangle, and a trapezoid, and then add up the areas of the four shapes to get the total area. Created by Sal Khan.

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Video transcript

- [Instructor] So we have the strange-looking shape here, and then were given some of its dimensions. We know that this side right over here has a length of 3.5, this side over here is 6.5, then we know from here to here is two, and then from here to here is seven, and then they're giving us this dimension right over here is 3.5. And so given that, let's see if we can find the area of this entire figure. And I encourage you to pause the video right now and try this on your own. So I assume you've given a go at it and there might be a few things that jump out at you immediately. The first thing is that they have these two triangles up here and they give us all of the dimensions for 'em, or at least they give us the base and the height for it, which is enough to figure out the area. If I had a triangle or if I had a rectangle that was 2 X 3.5 or two wide, two units wide and 3.5 units high. If had a rectangle like that, we know that it would have an area of 2 X 3.5. Now a triangle is just going to be, or especially a triangle like this, a right triangle is just gonna be half of a rectangle like this. We just care about half of its area. So this area is going to be 1/2 X 2 X 3.5. And 1/2 X 2 = 1, 1 X 3.5 is 3.5 square units. So the area of that part is going to be 3.5 square units. Now let's think about the area of this triangle right over here. Well, once again, we have its height as 3.5, its base is seven. So its area is going to be 1/2 X 7 X 3.5. 1/2 X 7 is 3.5 X 3.5, so this part is 3.5, and then I'm gonna multiply that times 3.5 again. So let's figure out what that product is equal to. So 3.5 X 3.5. 5 X 5 is 25. 3 X 5 is 15 + 2 is 17. Let's cross that out, move one place over to the left. 3 X 5 is 15. 3 X 3 is 9 + 1 is 10. So that gets us to 5 + 0 is five, 7 + 5 is 12. Carry the one, 1 + 1 is two, and we have a one. And we have two digits to the right of the decimal. One, two, so we're gonna have two digits to the right of the decimal in the answer. So the area here is 12.25 square units. Now this region might seem a little bit more maybe, maybe a little bit more difficult because it's kind of a, it's this weird trapezoid-looking thing. But one thing that might pop out at you is that you can divide it very easily into a rectangle and a triangle. And we can actually figure out the dimensions that we need to figure out the areas of each of these. We know what the width of this rectangle is, right? Or the length of this rectangle, whatever you want to call it. It's going to be two units plus seven units. So this is going to be nine. We know that this distance is 3.5. So if this distance right over here is 3.5, then this distance down here has to add up with 3.5 to 6.5. So this must be three. So now we can actually figure out the area. So the area of this rectangle is just going to be its height times its length, or 9 X 3.5. 9 X 3.5. And one way you could do it, we could even try to do this in our head. This is gonna be 9 X 3 + 9 X 0.5. So 9 X 3 is 27. 9 X 0.5, that's just half of nine, so it's gonna be 4.5. 27 + 4 would get us to 31. So that's going to be equal to 31.5. Or you could multiply it out like this if you like. But the area of this region is 31.5. And then the area of this triangle right over here, it's gonna be 9 X 3 X 1/2, we're looking at a triangle. 9 X 3 is 27, 27 X 1/2 is 13.5. And so to find the area of the entire thing, we just have to sum up these areas. So we have 31.5, 31.5 + 13.5, 13.5, plus 12.25, 12.25, plus 3.5, plus 3.5 here. So we just have a five here in the hundredths. That's the only one. 5 + 5 is 10 + 7 is 17. 1 + 1 is 2 + 3 is 5 + 2 is 7 + 3 is 10. 1 + 3 is 4 + 1 is 5 + 1 is six. So we get a total area for this figure of 60.75 square units.