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### Course: 7th grade > Unit 9

Lesson 2: Area and circumference challenge problems# Impact of increasing the radius

If we change the radius of a circle, how does the circumference and area change? Created by Sal Khan.

## Want to join the conversation?

- Is there a video on

and*Circumference*

? I don't know how to do that math!**Rotation**(12 votes)- Nevermind! I understand now.

But if any of you are wondering that:*The amount of times the circumference rotated, when you are given these circumstances is:***Diameter is 8:**`8*pi = circumference, then circumference * the amount of rotations`

Radius is 4:`(4*2)pi = circumference, then circumference * the amount of rotations`

`Circumference is 25.1327412287 then multiply that by the amount of rotations which is 10`

=251.327412287(11 votes)

- 100 - 4 (3^2) pi? doesn't that give us all four sides? I'm confused(10 votes)
- Hi! I think you meant to post this on the previous video (Area of a shaded region), right? :)(10 votes)

- Hi, what happens if the radius is tripled? How does it affect the area?(7 votes)
- That's an interesting question - if you expand it to three dimensions (what happens to the area and the volume if an object's size increases) it leads to the square-cube law: http://en.wikipedia.org/wiki/Square-cube_law

In two dimensions the answer is simply: as the radius increases, the area increases by the square of the increasing factor.

That's a bit abstract, so let's give some examples.

If the radius doubles, that's a factor of 2. The square of 2 is 4, so if the radius doubles the area is multiplied by 4. This is the example given in the video.

If the radius triples, that's a factor of 3. The square of 3 is 9, so if the radius triples the area is multiplied by 9.

You can see this goes up rather quickly: multiply the radius by 5, and the area is multiplied by 25. If the radius is multiplied by 100, the area is multiplied by 10000, etc.(3 votes)

- I could not understand how 2x^2 is equal to 4x^2....(5 votes)
- Throwing this out as a correction to akshith's comment. (2X)²
**IS**in fact equal to 4X².

breaks down into`(2X)²`

Which is the same thing as`2X ⋅ 2X`

You can use the commutative principle at this point to switch things around:`2 ⋅ X ⋅ 2 ⋅ X`

You can multiply both 2's to collapse this down to`2 ⋅ 2 ⋅ X ⋅ X`

And finally Both X's multiplied together can collapse down to`4 ⋅ X ⋅ X`

So you now end up with`4 ⋅ X²`

You can try testing both statements with a real value for X. For example, setting X = 3. If you do this and solve, you would see that (2⋅3)² = 4⋅3². Make sure you pay attention to the order of operations here.`4X²`

(9 votes)

- Why does sal say that A=Pi(2x)power of two= A=Pi(4x)Power of two? When squaring 2x to 4x, don't you need to take away the to the power of two? Sal's equation is saying you need to square it after you've already squared it. Am I missing something? Because I don't understand how that works.(5 votes)
- Woah!
`A = pi*(2x)^2`

The`(2x)^2`

actually turns into`4x^2`

. Why? The squared is to the right of the brackets, so it applies to**both**the factors`2`

and`x`

. To put it in another way, imagine you**distributed**the`^2`

to each of the factors.

It's the property where`(ab)^n = a^n*b^n`

.`(2)^2 = 4`

and`(x)^2 = x^2`

So,`A = pi*4x^2`

.

I hope it cleared things up.(1 vote)

- I'm confused about why both of the areas have increased by a factor of 4. The areas are pie x squared and four pie x squared(3 votes)
- When we are finding the area of an object, everything will have to be squared (That's why your answer has an exponent of 2 on the unit. It's two-dimensional). If you double the radius of a circle, its area will quadruple, because you must multiply by the factors in which you increased or decreased the side length(s), or in this case, the radius. Thus you would do 2x2, which would give you four. The reason we use two 2s is that in a circle, the radius is the length from the center of the circle to any given point on its circumference. If you were doing this for rectangles or squares, or many other shapes, you can utilize the same method.

For example, we have a rectangle with lengths of 7 and a width of 3. Say we decide to double the length and quadruple the width. How many times bigger than the original will the new rectangle be? Well, all you would need to do is multiply 2 by 4. Your resulting answer would be 8. Now, to verify this, you can do it the long way. First you multiply 3 by 7, and end up with the answer 21. Now, you must do (2x7)x(4x3). Your answer would be 168. Now divide 168 by 21. What's your answer? Eight. You would use the same algorithm if you were shrinking the shape.

Well, I hope this helped. Be sure to let know if you need any clarification on my answer! I'm here to help. Until then, peace! :D(4 votes)

- By how many units must the radius of a circle be increased to increase its

circumference by 22p units?(3 votes)- The radius must be 11 units for the circumference to be 22 pi units since the formula for circumference is 2*pi*radius. Hope that helps! :):)(3 votes)

- By how many units must the radius of a circle be increased to increase its

circumference by 22p units?(4 votes) - As someone noted below, Pi*D (diameter) is the same as pi*2R. Is there a reason to go with one rather than the other? I ask because pi*2R or 2piR whichever way, sounds similar to area pi*R^2. Granted if you're paying attention it shouldn't matter, but it seems like it's easier to keep them straight in memory using pi*D especially since Pi is defined by diameter--at least in the wording of the definition "Pi is the ratio of the circumference to the diameter."

Any reason to use Pi*2R over Pi*D?(3 votes)- You can choose which way you express the circumference of a circle. The reason that most people use 2πr as the formula is because the radius of a circle or a face of a circular object is used more often than the diameter. For example, the area of a circle is πr^2. This is much easier than writing π(d/2)^2.(3 votes)

- how many units must the radius of a circle be increased to increase its circumference by 22p units?(4 votes)

## Video transcript

- [Voiceover] What I
want to do in this video is think about "How does the circumference "and how does the area of a circle change "as we change its radius?" And, in particular, we'll focus on what happens when we double its radius. Let's think about a
circle right over here. So, this is a circle. And let's say its radius is X units. Whatever our units is. This distance right over here is X. And then let's think about another circle that has twice the radius. Its radius will be two X. Let me draw its radius first so it looks roughly accurate. This is two X is this circle's radius. And so this circle might
look something like that. That's my best attempt at
freehand drawing a circle. Let's just think about what the circumference
of both of these are, and what the areas of both of these are. The circumference of any circle is two pi times the radius. So, in this case, the circumference, and I'll use C for circumference, is equal to two pi times the radius. Which, in this case, is X. What's the circumference here? Once again, the circumference is equal to two pi times the radius, but this time the radius is two X. So, the circumference is equal to two times pi times two times X, which is the same thing as two times two times pi times X, or we could write it as four pi X. We see here that this circumference is twice as large as this one. To go from two pi X, to four pi X, you have to multiply by two. You double the radius, it doubled the circumference. What about the area? And I'll do area in a new color. We already know that area
is equal to pi R squared. In this circle the radius is of length X. It's pi times X squared. In this circle, right over here, the area is going to be equal to pi times the radius squared, but now the radius is two X. Two X squared. What is this going to be equal to? Our area is equal to pi. Two X squared is two X times two X, which is the same thing as four X squared. Four X squared. Or, we could rewrite this as area is equal to four pi X squared. Notice, now the area has increased not by a factor of two. The area has increased by a factor of four when
we doubled the radius. Why did this happen? I encourage you to pause the video and think about it. It comes straight out the formulas for circumference and area. Remember, circumference
is equal to two pi R, while area... Let me do this in a different color. While area is equal to pi R squared. You see here, area is proportional to the square of the radius. If you double this, you're gonna increase your
area by a factor of four. If you triple it, if
you triple your radius, you're gonna increase your
area by a factor of nine. If you increase radius
by a factor of four, you're gonna increase your area by a factor of four squared, or 16. While, circumference, whatever factor you increase
you increase your radius, the circumference is going to
increase by that same factor. And if you don't believe me, I mean, we essentially
showed it right here through a little bit of algebra, but you could try it out with as many numbers as you see fit.