If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

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?

• 100 - 4 (3^2) pi? doesn't that give us all four sides? I'm confused • Hi, what happens if the radius is tripled? How does it affect the area? • 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.
• Is there a video on `Circumference` and `Rotation`? I don't know how to do that math! • 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`
`(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
• Math makes me want to cry • I could not understand how 2x^2 is equal to 4x^2.... • Throwing this out as a correction to akshith's comment. (2X)² IS in fact equal to 4X².
``(2X)²``
breaks down into
``2X ⋅ 2X``
Which is the same thing as
``2 ⋅ X ⋅ 2 ⋅ X``
You can use the commutative principle at this point to switch things around:
``2 ⋅ 2 ⋅ X ⋅ X``
You can multiply both 2's to collapse this down to
``4 ⋅ X ⋅ X``
And finally Both X's multiplied together can collapse down to
``4 ⋅ X²``
So you now end up with
``4X²``
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.
• Amogus • 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 • 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
• By how many units must the radius of a circle be increased to increase its
circumference by 22p units?   