Main content

### Course: 6th grade (Illustrative Mathematics) > Unit 1

Lesson 1: Lesson 3: Reasoning to find area# Finding area by rearranging parts

Discover the magic of quadrilaterals and their areas! We explore how a trapezoid's area can match a rectangle's by rearranging its parts. We learn that shapes with the same area can look different. It's a fantastic journey into the world of geometry! Created by Sal Khan.

## Want to join the conversation?

- i dont understand any of this(58 votes)
- What don't you understand? If its why moving the shape works, then its because the area of the shape is only being shifted to another side of the object. This still counts as part of the area even though its shifted. The reason why you shift part of the are to another side is so that its easier to calculate.(42 votes)

- what im still confused i dont understand anything.(20 votes)
- well, they are showing how these different quadrilateral's area is being found. The green one can either be like blue(if you remove the extra triangles) or the red(if you finish the triangle by bringing up or down one.) and for the pink we have an extra that cant be finished, so it is left, and if you compare green and pink pink is bigger with the extra.

hope it is useful!(12 votes)

- I have math homework that is about finding the area of a polygon, could you make a video on that please ?

thank you(17 votes)- Finding the area of a polygon is quite simple. I'll explain how to do so here. Let's say we have a hexagon. Each side of the hexagon is 7 inches. The first step is to divide the hexagon into triangles and find one of their areas. Say the triangle had a base of 7in and a height of 4in. Figure out the area (which would be 14 sq in.) Now, multiply 14 by the amount of sides that the shape has. So 14x6, which is 84. Therefore, the area of the polygon is 84 square inches.

I hope this helped!(22 votes)

- This got super easy really quick!

So if i can do it you can too! :D(21 votes) - With the pink one, why did he not count the 'extra'?(10 votes)
- Sal was desperately in search for the one similar to the green quadrilateral . So , after finding out that the green one had extra right angled triangle , he skipped that and went forward . Hope that helped.(21 votes)

- how is that supposed to answer my problem?(18 votes)
- I love listening to this guy. he has such a calm voice, like Bob Ross(13 votes)
- i am so confused i know what to do but at the same time i do not how to have a shape that does not already have the base and height for me(10 votes)
- brake the shape down into other shapes and then move them to make a parelelagram that is in the vidio(3 votes)

- why so fat like why(11 votes)
- Why do they always have to teach us the more complicated method?(8 votes)

## Video transcript

- [Instructor] We have four quadrilaterals drawn right over here. And what I want us to think about, is looking at this green
quadrilateral here, I want you to pause the video and think about which of these figures have the same area as
the green quadrilateral. And so pause the video
now and think about that. So I'm assuming you gave a shot at it. Now let's think about it. And the way I'm gonna think about it is to really rearrange parts
of this green quadrilateral to make it look more like maybe some of these other quadrilaterals. So for example, if we were to put a little dotted line right over here and a dotted line right over here, we see that our green
shape is actually made up, you could imagine it being
made up of a triangle, and then a rectangle, and
then another triangle. And what's interesting
about the two triangles is that they represent
the exact same area. They essentially each represent half of this rectangle right over here. They represent half of
this rectangle down here. Let me do that in a color. They represent half of this entire thing. If I were to color it all in. And if you have trouble visualizing it, imagine taking this top
part right over here and then flipping it over,
it would look like this. If you flip it over this
line right over here, it would look something like this. It would look something like this. My best attempt to draw it. So take that top section, it
would look something like that, and then move it down and then move it down right
over here to fit in here. And then this, plus this will fill in this entire region right over here. So that original green trapezoid
that we were looking at, if you take that top part out, it essentially has the exact
same area as a rectangle that has a height of four
and a length of five. So this right over here has the exact same area as our trapezoid. And once again, how did we do that? Well, we just took this
top part, flipped it over, and relocated it down here, and we said, hey, we
could actually construct a rectangle that way. So essentially, and if
you wanna know its area, we could either just
count the squares here. So we have, let me do
this in a easier to see. So we have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 of these unit squares right over here. And we know that there's
an easier way to do that. We could have just multiplied
the height times the width. We could have just said, look,
this thing is 1, 2, 3, 4 high and 1, 2, 3, 4, 5 wide. So four times five is gonna give us 20 of these unit squares. So that's the area in
terms of unit squares or square units of that
original green trapezoid. Now let's see which one
of these match that. So this pink one right over here, if you don't even count this bottom part, if you were to just separate
this top part right over here, this top part is four high by five wide. So just this top part alone is 20. And then it has this
extra right over here. So the pink has a larger area than our original green trapezoid. The blue rectangle is three by five. So it has an area of 15 square units. Now the red one is interesting. It is 1, 2, 3, 4 high and 1, 2, 3, 4, 5 long or five wide. Four times five is 20 squares. And you can validate that. And so the red rectangle has the same area as our original green trapezoid.