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so we have a circle here and let's say that we know that its circumference is equal to 6 pi I'll write it unit whatever our units happen to be let's see if we can figure out given that its circumference is 6 pi of these units what is the area going to be equal to pause this video and see if you can figure it out on your own and first think about if you could figure out the area for this particular circle and then let's see if we can come up with a formula for given any circumference can we figure out the area and vice versa all right now let's work through this together the key here is to realize that from circumference you can figure out radius and then from radius you can figure out area so we know that circumference which is 6 pi so we know 6 pi is equal to 2 pi times our radius and so what is the radius going to be so the radius we're talking about that distance well we can divide both sides by 2 pi so let's do that if we divide both sides by 2 pi to solve for our what are we left with well we have an R on the right hand side we have R is equal to pi divided by pi that's just 1 6 divided by 2 is 3 so we get that our radius right over here is equal to 3 units and then we can use the fact that area is equal to PI R squared to figure out the area this is going to be equal to PI times 3 squared I don't have to write parenthesis there pi times 3 squared which is of course going to be equal to 9 pi so for this particular example when the circumference is 6 pi units we're able to figure out that the area this is actually going to be 9 pi square units or I could write units squared because we're squaring the radius the radius is 3 units so you square that you get the units squared now let's see if we can come up with a general formula so we know that circumference is equal to 2 PI R and we know that area area is equal to PI r-squared can we come up with an expression or a formula that relates directly between circumference and area and I'll give you a hint solve for you could solve for R right over here and substitute back into this equation or vice versa pause the video see if you can do that alright so let's do it over here let's solve for R if we divide both sides by 2 pi do it another color so if we divide both sides by 2 pi and this is exactly what we did up here what are we left with we're left with on the right hand side R is equal to C the circumference divided by 2 pi the radius is equal to the circumference over 2 pi and so when we're figuring out the area area remember is equal to pi times our radius squared but we know that our radius could be written as circumference divided by 2 pi so instead of radius I'll write circumference over 2 pi remember we want to relate between area and circumference and so what is this going to be equal to we get area is equal to PI times circumference squared over 2 pi squared is 4 pi squared let's see we have a PI or we would have if we multiply this out we'd have a pi in the numerator and a PI into the NAM or 2 PI's in the denominator being multiplied so pi divided by pi squared is just 1 over pi and so there you have it area is equal to circumference squared divided by divided by 4 pi let me write that down so this is a neat you don't tend to learn this formula but it's cool that we were able to derive it area is equal to circumference squared over 4 pi and we can go the other way around give it an area how do we figure out circumference we could just put the numbers in here or you could just solve for C let's multiply both sides by 4 pi let's multiply both sides by 4 pi and if we do that what do we get we would get 4 pi times the area is equal to our circumference squared and then to solve for the circumference we just take the square root of both sides so you would get the square root of four pi times the area is equal to our circumference and you could simplify this a little bit if you wanted you could take the four out of the radical but this is pretty neat how you can relate circumference and area