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### Course: 3rd grade>Unit 10

Lesson 5: Area and the distributive property

# Area and the distributive property

Sal uses the distributive property to find area of rectangles. Created by Sal Khan.

## Want to join the conversation?

• why do they call it distributive property?
(18 votes)
• Because the term outside the parentheses is distributed to all the terms inside the parentheses. Distribute is another way to saying share with.
(28 votes)
• At it says the numbers are equivalent ,how is that?
(16 votes)
• As I understand, it means that both formulas will give you the same number; so it doesn't matter how you go about finding the area, either with formula a or b, both will provide the same result and therefore are equivalent.
(20 votes)
• wut is money like bro how do I count it like how
(1 vote)
• A wire is in the shape of a square of perimeter 64 cm. A rectangle is formed with this wire such that its length is 20 cm. then,the area of the rectangle is____?
(1 vote)
• From the perimeter of the square, we know that the length of the wire is 64 cm, and so the perimeter of the rectangle will also be 64 cm.
Perimeter of rectangle = 2 * (length + width)
and so 64 = 2 * (length +width)
to get the sum of of the length of width, we divided 64 by 2 and get 32.
We know the length is 20 cm. To find the width, we subtract 20 from the sum of the length and the width, which is 32. 32-20 = 12 cm
Now we know the length and the width.
Area = 12 * 20 = 240 cm squared
(1 vote)
• A rectangle has a length of 28cm and breadth of 12cm. What is the area of the least numbers of these rectangles used to form the smallest square possible?
(1 vote)
• That's an interesting problem! My first thought is to find the factors of the sidelengths so that we can work out a square shape to make if we simply placed them side by side. By finding the factors we can find the lowest common multiple of the two numbers.

``12 = 2 x 2 x 328 = 2 x 2 x 72 x 2 x 3 x 7 = 84``

So if we placed them side by side we would get an 84cm by 84cm square. This would involve 21 rectangles, with 3 rows of 7, as 84/12 = 7 and 84/28 = 3. So the area will be 84 x 84 which equals 7056cm^2

You might like to try videos from this section for more help:

https://www.khanacademy.org/math/pre-algebra/factors-multiples/greatest_common_divisor/v/lcm-and-gcf-greatest-common-factor-word-problems
(1 vote)
• At , what are the two strategies?
(1 vote)
• As he says around , the two different strategies or ways to do this would be to find the area of each small rectangle and add them together or just find the area of the whole rectangle by multiplying the entire width by the entire length.

For the first way, you find the area of the two rectangles separately. The area of the first small rectangle is 9 x 8 or 72. The area of the second rectangle is 9 x 12 or 108. Then you add the two areas together (as the two small rectangles makes up the big one) to get 180 as the total area.

The second is just to take the entire width and multiply it by the entire length to get the area of the big rectangle. This would be 9 x (8 + 12) which is 9 x 20 or 180.

Either way, you get the area of the total rectangle as 180. Later on you can prove that 9 x (8 + 12) = 9 x 8 + 9 x 12, through the distributive property.
(1 vote)
• Could you use the distributive property for something else other than area?
(1 vote)
• Yes. Every time you see a number (or variable) multiplying a bracket inside which there is an addition, that's when you can use the distributive property. It's one of the rules of algebra, not just area, so you can use it in any situation you need to.
(0 votes)
• Is the x supposed to be a variable or a multiplication sign because it is very confusing.
(0 votes)
• This type of confusion is the reason why, in algebra and higher levels of math, other symbols instead of x are used for multiplication (such as the asterisk symbol *, the dot symbol, and parentheses). Use of these other symbols for multiplication avoids this confusion.

Have a wonderful day!
(2 votes)
• So, just to make sure I'm not being "tricked" here somehow... We are obliged to use the "distributive property" to make sure that the "order of operation" is being applied, right?! I mean, multiplication before addition, right? ^_^
(0 votes)
• Not exactly. According to the order of operations (PEMDAS) you must evaluate the terms in the parentheses first. He was not saying that you must use the distributive property to calculate the area, just that this problem serves as an example of why the distributive property is true.

9(8+12) == 72+108 == 9(20) == 180

Since this is essentially a simple addition and multiplication problem, the Commutative Property dictates that the order in which the terms are evaluated will not affect the answer.
(2 votes)
• why do they call it distributive property?
(0 votes)
• They call it the distributive property because....

(example)

5(2+6)
For something like this, you distribute 5 to 2 and 5 to 6.

Like this:

(5*2) + (5*6)
In this you distributed the 5 to the 2 and 6

So really the word means to distribute numbers to numbers.

Hoped this helped! :D
(0 votes)

## Video transcript

I have this rectangle here, and I want to figure out its area. I want to figure out how much space it is taking up on my screen right over here. And I encourage you to pause this video and try to figure out the area of this entire rectangle. And when you do it, think about the two different ways you could do it. You could either multiply the length of the rectangle times the entire width, so just figure out the area of the entire thing. Or you could separately figure out the area of this red or this purple rectangle and then separately figure out this blue rectangle and realize that their combined areas is the exact same thing as the entire rectangle. So I encourage you to pause the video and try both of those strategies out. So let's just try them out ourselves. First, let's look at the overall dimensions of the larger rectangle. The length is 9, and we're going to multiply that times the width. But what's width here? Well, the width is going to be 8 plus 12. This entire distance right over here is 8 plus 12. So it's 9 times 8 plus 12. This is one way that we could figure out the area of this entire thing. This is just the length times the width. 8 plus 12 is obviously going to be equal to 20. But the other way that we could do it-- and this must be equivalent, because we're figuring out the area of the same thing-- is to separate out the area of these two sub-rectangles. So let's do that. And this must be equal to this thing. So what's the area of this purple rectangle? Well, it's going to be the length. It's going to be 9. Let me do it in that same color. It's going to be 9 times the width, which is 8. It's going to be 9 times 8. And then what's the area of this the blue rectangle? Well, that's going to be 9 times-- so the height here is 9 still. The height is 9. And what's the width? Well, the width is 12. And what's the area of the combined if you wanted to combine the area of the purple rectangle and the blue one? Well, you'd just add these two things together. And of course, when you add these two things together, you get the area of the entire thing. So these things must be equivalent. They are calculating the same area. Now, what's neat about this is we just showed ourselves the distributive property when we're dealing with these numbers. You could try these out for any numbers. They'll work for any numbers, because the distributive property works for any numbers. You see 9 times the sum of 8 plus 12 is equal to 9 times 8 plus 9 times 12. We essentially have distributed the 9-- 9 times 8 plus 9 times 12. And let's actually calculate it just to satisfy ourselves about the area. So if you multiply the length times the entire width, so that's 9 times 8 plus 12, that's the same thing as 9 times 20, which is 180. And over here, if you calculate the area of this purple rectangle, that is 9 times 8. So that is going to be equal to 72. That's the purple rectangle. The area of the blue rectangle, 9 times 12, well, that's 108. And we're going to take the sum of the two to find the area of the larger rectangle. What is 72 plus 108? Well, 72 plus 108 is also equal to 180. So we've verified. These, indeed, are equal to each other when we calculate them. And they make sense, because we are calculating the same exact area.