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Current time:0:00Total duration:3:31

CCSS.Math:

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 so 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 so this distance right over here is X and then let's think about another circle that has twice the radius so its radius will be 2x and so let me draw its radius first so I get so it looks roughly or accurate so this is 2x is this circles radius and so the circle might look something like that that's my best attempt at freehand drawing a circle so let's just think about what the circumference of both of these are and what the areas of both of these are so the circumference of any circle is 2 pi times the radius so in this case their circumference and I'll use C for circumference is equal to 2 pi times the radius which in this case is X well what's the circumference here well once again the circumference is equal to 2 pi times the radius but this time the radius is 2x so the circumference is equal to 2 times pi times 2 times X which is the same thing as 2 times 2 times pi times X or we could write it as 4 pi X so we see here that this circumference is twice as large as this one to go from 2 pi X to 4 PI X you have to multiply multiplied by 2 so you double the radius it doubled their circumference now what about the area and I'll do area in a new color so we already know that area is equal to PI R squared in this circle the radius is of length x so 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 2x 2x squared now what is this going to be equal to well our area is equal to PI 2x squared is 2x times 2x which is the same thing as 4 x squared 4 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 a factor of four when we doubled the radius now why did this happen and encourage you to pause the video and think about it well it comes straight out of the formulas for circumference in area remember circumference is equal to two PI R while area while area I'm just in a different color while area area is equal to PI R squared so you see here areas of proportional to the square of the radius so if you double this you're going to increase your area by a factor four if you triple it if you triple your range you're going to increase your area by a factor of nine if you increase your radius by a factor of four you're going to increase your area by a factor of four squared or 16 while circumference whatever factor 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