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### Course: Math (NSDC) - English>Unit 11

Lesson 2: Area of rectangles

# Counting unit squares to find area formula

Sal uses unit squares to see why multiplying side-lengths can also find the area of rectangles.    Created by Sal Khan.

## Want to join the conversation?

• Could I use this information for inches as well?
• Yeah. If you do the same thing, but change the measurement, you can do the same thing for inches like you did with meters.
• What if the units don't fit in the real shape?
• If the units don't fit exactly, then you need to use fractions of the unit. For example, if I draw a straight line and it is more than three inches long, but shorter than four inches, then it would be 3 and something inches long (for example 3 1/2 inches).
• What is a meter?
• A meter is a unit of measurement in the metric system it is about 39 inches, so a little longer than a yard.
• Does it matter if it's a rectangle or a square in area? Wouldn't it still be length times width?
• That's true. The only thing is, when calculating the area of a square, you can simply multiply one of the sides by itself, since you know the other sides are all the same.
• can i use this information for finding area in triangles?
• No, you use area by multiplying base times height, then divide by 2 by using square units.
• How is a square a rectangle?
• A rectangle is a quadrilateral (A shape with 4 sides) in which all 4 angles are right angles; opposite sides parallel and equal.

A square meets all of these criteria, thus a square is also a rectangle.
• If a rectangle has width of 0.5cm and length of 0.6cm does the area equal to 0.3cm^2? and if it's true, how is this area less than width and length?
• Yes. That's right and thought-provoking. You can't really compare the length or width to the area. One is cm and the other is cm^2. It's impossible to say if a pencil is longer than an hour. It's asking you to compare time to length, which isn't possible. The same thing happens in this case.
• I'm a Chinese-speaker student. At , I can't understand the sentence that "Just fill it really good about multiplying the dimensions of this rectangles." Thank you who might help me overcome this language hatch very much! Although it's hard for me to watch videos at Khanacademy. org in English, I've gained a lot of fun and knowledge that I've never acquired in my classroom in China.
• What he said was "just to feel really good about multiplying the dimensions of these rectangles." This means that he wants to make sure we are comfortable multiplying the dimensions of rectangles, or he wants to make sure we know how to do it, why to do it, and why it works.

I hope this helps!
• When we're doing this, how do you find the area if one of the perimeters is a variable, and you have to figure out the variable first?
• Well, first there are two terms that sound similar but have different meanings. The perimeter of a shape ( or polygon) is the distance around the edge. A parameter is part of a function (or formula), similar to a variable, but it is held constant for a given problem.

So if you are asked to find the area of a rectangle and you are given the length as a measurement, say 5m, and you are given the width as a variable, say W, then you can still multiply these two values to get: Area = 5 x W = 5W square meters. Once you find out what W is, you can plug it in to the equation to find the area.
• Why is the square unit in the middle counted?

ex.
3*3=9 3 rows and 3 in each row
• The area (unlike the perimeter) of a two-dimensional figure can be thought of as the amount of space inside the figure. The middle square is still inside the figure, so it is counted in the area.

## Video transcript

I've got three rectangles here, and I also have their dimensions. I have their height and their width. And in fact, this one right here has the same height and width, so this is actually a square. So let's think about how much space they each take up on my screen. And since we're doing everything in terms of meters, since all of the dimensions are in meters, I'm going to measure the area in terms of square meters. So let's see how many square meters I can fit onto this yellow rectangle without going outside of its boundary and without overlapping. So I can fit 1 square meter. Remember, a square meter is just a square where its length is 1 meter and its width is 1 meter. So that's 1 square meter, 2, 3, 4, or 5, and 6 square meters. So we see here that the area is 6 square meters. Area is equal to 6 square meters. But something might be jumping out at you. Did I really have to sit and count 1, 2, 3, 4, 5, 6? You might have recognized that I could view this as really 2 groups of 3. And let me make that very clear. So, for example, I could view this as one group of 3 and then another group of 3. Now, how did I get groups of 3? Well, that's because width here is 3 meters. So I could put 3 square meters side by side. And how did I get the 2 groups? Well, this has a length of 2 meters. So another way that I could have essentially counted these six things is I could have said, look, I have a length of 2 meters. So I'm going to have 2 groups of 3. So I could multiply 2 times 3, 2 of my groups of 3, and I would have gotten 6. And you might say, hey, wait. Is this just a coincidence that if I took the length and I multiplied it by the width, that I get the same thing as its area? And no, it's not, because when you took the length, you essentially said, well, how many rows do I have? And then you say when you multiply it by the width, you're saying, well, how many of these square meters can I fit into a row? So this is really a quick way of counting how many of these square meters you have. So you could say that 2 meters multiplied by 3 meters is equal to 6 square meters. Now, you might say, hey, I'm not sure if that always applies. Let's see if it applies to these other rectangles right over here. So based on what we just saw, let's take the length, 4 meters, and multiply by the width, and multiply by 2 meters. Now, 4 times 2 is 8. So this should give us 8 square meters. Let's see if this is actually the case. So 1, 2, 3, 4, 5-- and you see it's going in the right direction-- 6, 7, and 8. So the area of this rectangle is, indeed, 8 square meters. And you could view this as 4 groups of 2. So you could literally view this as 4 groups of 2. That's where the 4 times 2 comes from. So you could view it as 4 groups of 2 like this. Or you could view it as 2 groups of 4, So 1 group of 4 right over here. So you could view this is 2 times 4, and then 2 groups 4. I want to draw it a little bit cleaner. Now, you could probably figure out what the area of this rectangle is. It's actually a square, because it has the same length and the same width. We multiply the length, 3 meters, times the width, so times 3 meters, to get 3 times 3 is 9-- 9 square meters. And let's verify it again just to feel really good about this multiplying the dimensions of these rectangles. So we have 1, 2, 3, 4, 5, 6, 7, 8, and 9. So it matches up. We figure out how many square meters can we cover this thing with, without overlapping, without going over the boundaries. We get the exact same thing as if we multiplied 3 times 3, if we multiplied the length times the width in meters.