Main content

## 6th grade (Illustrative Mathematics)

### Unit 1: Lesson 6

Lesson 11: Polygons# Area of composite shapes

CCSS.Math:

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.

## Video transcript

We have this strange
looking shape here, and then we're 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 2, and then from here to here is 7. And then they're giving us
this dimension right over here is 3.5. 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. 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 them, 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 rectangle that was 2
units wide and 3.5 units high, if we know that it would
have an area of 2 times 3.5. Now a triangle is
just going to be, especially a triangle like
this, a right triangle, is just going to be half
of a rectangle like this. We just care about
half of its area. So this area is going to
be 1/2 times 2 times 3.5. 1/2 times 2 is equal to 1. 1 times 3.5 is 3.5 square units. So the area of that part is
going to be 3.5 square units. Let's think about the area of
this triangle right over here. Well, once again we
have its height is 3.5. Its base is 7. So its area is going to
be 1/2 times 7 times 3.5. 1/2 times 7 is 3.5 times 3.5. So this part is
3.5, and I'm going to multiply that
times 3.5 again. Let's figure out what
that product is equal to. 3.5 times 3.5. 5 times 5 is 25. 3 times 5 is 15, plus 2 is 17. Let's cross that out. Move one place over to the left. 3 times 5 is 15. 3 times 3 is 9, plus 1 is 10. So that gets us
to 5 plus 0 is 5. 7 plus 5 is 12, carry the 1. 1 plus 1 is 2. And we have a 1. We have two digits to the
right of the decimal, one, two. So we're going to
have two digits to the right of the
decimal in the answer. The area here is
12.25 square units. Now this region may be a
little bit more difficult, because it's kind of us 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, or the length of
this rectangle, whatever you want to call it. It's going to be 2
units plus 7 units. So this is going to be 9. We know that this
distance is 3.5. 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 3. Now we can actually
figure out the area. The area of this
rectangle is just going to be its height times
its length, or 9 times 3.5. 9 times 3.5. And one way you could
do it-- we could even try to do this in our head--
this is going to be 9 times 3 plus 9 times 0.5. 9 times 3 is 27. 9 times 0.5, that's just half of
nine, so it's going to be 4.5. 27 plus 4 will 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 is going to be 9
times 3 times 1/2. We're looking at a triangle. 9 times 3 is 27. 27 times 1/2 is 13.5. So to find the area
of the entire thing, we just have to
sum up these areas. We have 31.5 plus 13.5
plus 12.25 plus 3.5. So we just have a 5
here in the hundredths. That's the only one. 5 plus 5 is 10, plus 7 is 17. 1 plus 1 is 2, plus 3 is 5,
plus 2 is 7, plus 3 is 10. 1 plus 3 is 4, plus
1 is 5, plus 1 is 6. So we get a total area for this
figure of 60.75 square units.