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Area of a circle intuition

CCSS.Math:

Video transcript

what I'd like to do in this video is make an informal argument for why the formula for the area of a circle is PI R squared and we're going to start just with the most traditional definition of the number pi and that's that and I'll just do it here in a corner someplace that pi is equal to the ratio of the circumference and the diameter of the circle or the ratio of the circumference to the diameter of a circle or of course we could write this as the ratio of the circumference to instead of the diameter I could write 2 times the radius or I could multiply both sides times 2 times the radius and we get our traditional formula for the circumference for the circumference of a circle but once again this literally just comes straight out of the definition for the number pi the number pi is defined as the ratio between the circumference in the diameter so you just multiply both sides of it times the diameter you get you get the circumference is equal to PI times the diameter now we have the circumference formula right here once again this comes out of the definition of pi but from this I'd like to at least get an intuitive feel for why the area formula is given by PI R squared and to think about that we're going to approximate the area of polygons the areas of polygons that are inscribed in a circle so over here I have this was this was a five-sided polygon right over here and it's area is going to be equal to so the area of this polygon it's going to be 5 times the area of each of those triangles in the area of each of those triangles the height is a the base is B so it's going to be base times height or height times base times 1/2 so this once again 5 times a be over 2 it's not that good of an approximation this is this would be the area of just the inscribed polygon so we're definitely under estimating the area of the entire circle we're leaving out all of these little these little chunks outside of the polygon but still inside of the circle but as we add more size to the polygon we see that we're leaving out less we see when we have now this is a 1 2 3 4 5 6 seven sided polygons where we're leaving we're leaving a little bit less we're under estimating still but we're under estimating by less this area that we're giving up isn't as large as this area right over here so in this approximation we have would I say seven triangles one two three four five six seven triangles and the area of each of those triangles is once again a B over two now a and B here are different than a and B over here and notice what's happening as as we increase as we increase the number of triangles not only is it approximating the area of the circle better but a is getting longer and you can see as you could imagine as as we increase many many many more triangles a is going to approach are now another thing to think about is what is seven what is seven times B approaching so we're saying that a is approaching R as we add more of these sides of the polygon as we add more triangles and what is the number of triangles times the base of the triangle what is that approaching well this is going to approach is going to approach the perimeter of the pot or this is going to be the perimeter of the polygon so seven times B is that plus let me actually let me draw this it's that Plus that Plus that I think you get the point Plus that Plus that Plus that Plus that so once again seven let me write this as seven times B that is the perimeter of the polygon perimeter perimeter of the polygon so think about what's happening as we as we have more and more sides of the polygon are a are our height of each of our triangles is going to approach our radius is going to approach the radius it's going to get long the height of each of the triangle is gonna get longer and longer it's going to approach the radius as we have as we approach an infinite number of triangles and then the number of polygons we have or the number of plugins the number of sides we have times the basis that's going to be the perimeter of the polygon and as we add more and more sides as we add more and more size the perimeter of the polygon is going to approach is going to approach is going to approach the circumference of the circle I'll write it out sir cumference circumference and you see that even more clearly right over here so once again how many how many sides do I have here I have one two three four five six seven eight nine ten sides so this I can write the perimeter of the polygon as ten times B and then if I multiply that times a over two if I multiply that times just another color a over I'm just write it up like this times a over two I'm once again approximating the area of the circle because a times B over two that's the area of each of these triangles and then I have ten of these triangles but now let's think about this more generally let's think about it if if I were to have n if I would have an n-sided polygon so I have an n-sided polygon then I'd be approximating the area as n times B and times B we see this right over here when n is equal to 10 you have 10 times B so it's n times B times a over 2 times a over 2 and it's right why this isn't something mysterious the base times the height divided by 2 this is the this right over here that's the area of each triangle and then I'm going to have n of these triangles so this is our approximation for the area so let me write this the area is going to be approximately that right over there it's going to be n the number of triangles I have times the area of each triangle now what's going to happen as n approaches infinity is have it as I approach having an infinite sided on as I have an infinite number of triangles so let's just think this through a little bit because this is where it gets interesting and this is the informal argument to do this to do this better I'd have to I'd have to dig out a little bit of calculus but this gives you the essence so let's just think about what happens as n approaches infinity so as n approaches infinity we've already said as we have more and more sides we have more and more triangle eight more and more triangles a approaches are so let's write that down so a is going to approach R the height of the triangles is going to approach the radius and what else is going to happen well n times B the perimeter of the polygon the perimeter of the polygon is going to approach the circumference so a is going to approach R and n times B is going to approach the circumference is going to approach the circumference or another way of thinking about if it's approaching the circumference we could say that n times B is going to approach 2 pi times the radius because that's what the circumference is going to be equal to so if a is approaching the radius and n B is approaching 2 pi R well then what is the entire what is the area of what is the area of our polygon or what is the area of our of ours what is the area of our polygon going to or the area of our circle going to be well it is going to it is going to approach it is going or I should say the area of our polygon is going to approach and B is going to approach 2 PI R and B is going to approach 2 PI R so instead of n V I'm writing 2 pi are there a is going to approach our a is going to approach R and then I'm dividing it and then I am dividing it by 2 so as n approaches infinity is we have an infinite number of sides of our polygon an infinite number of triangles the area of our polygon will approach this which is equal to what well you have two divided by two and then PI R times R is equal to PI R squared so as our as we as we approach having an infinite number of triangles and infinite I want to keep that there an infinite number of sides we see that we approach the area of the circle and as we approach the area of the circle we are approaching PI R squared so hopefully this gives you an intuitive sense why this right over here is the formula for the area of a circle you could think about it is the area of an infinite site an infinite sided polygon that is inscribed in the circle which will be equal to the area of the circle