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Impact of increasing the radius

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

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.