Why is πr² the formula for a circle's area?

hey this is press tow Walker here's a formula we all learn in school a circle with the radius equal to R has an area that's equal to Pi * R 2 but have you ever wondered where does this formula come from and how was it possible that ancient civilizations had an understanding of this formula to dig deeper into this we have to go into the definition of Pi a circle with a circumference that's equal to C and a diameter that's equal to D which is equal to 2 R has a constant ratio between its circumference and diameter this was learned empirically by ancient civilizations you could imagine they were estimating the length of the circumference to this diameter different civilizations came up with different estimates one of the the earliest known estimates was by Archimedes and his value was approximately 3.14 ARA butut came up with the approximation of 3.1416 Lee way came up with the approximation of 3.14159 very interestingly these values were so accurate that they were not surpassed for nearly 1,000 years it turns out this ratio is exactly the approximation of what we call Pi but they themselves didn't use that symbol that came much later in approximately the year 1706 when William Jones used the symbol pi to denote exactly the ratio of the circumference to the diameter so while the Ancients were aware of this relationship they didn't exactly use our notation and the symbol pi has only been around for about 300 years so while you might find math class annoying today think about how cumbersome it was in ancient times without proper notation in any case returning to the point the definition of pi is the ratio of the circumference to the diameter which is approximately 3.14 because this is the definition we can rearrange the equation and we can get that the circumference is equal to Pi * the diameter and this is equivalent to 2 * < * R and the reason we need to know this is because when you're calculating the area of the circle in all of these derivations you need to know what the circumference of the circle is so we're going to end up needing to know that this circumference is equal to 2 pi r and that's by definition of Pi so now let's go over how the Ancients calculated the area of the circle I'll will present three different methods one method comes from Archimedes another method from Leonardo da Vinci and Sato motion and final method comes from Rabbi Abraham bar heia so let's get started with Archimedes so let's say we have a circle you can inscribe a regular hexagon in the circle and that will approximate the area but a neat fact about the regular hexagon is we can divide it up into triangles so let's say this is a side length s we can divide it up into six different equilateral triangles so let's calculate the area of a single triangle its height will be equal to a length a which we will call the apothem this is the distance from the center of this inscribed regular hexagon to this side so the area of this triangle will be equal to2 * its base s time the height a so 12 sa a in order order to get the area of the entire hexagon we need to multiply this by six so we have six multiplied by that area which can be Rewritten as 12 * 6 SAA but we know in this hexagon 6 * s will be equal to the perimeter of the hexagon so we can substitute that this is equal to 1/2 * the perimeter time a and this very formula will actually be true for every single regular polygon the area will be equal to2 * its perimeter time the apoem let's do another example with an octagon we will calculate the area of one of these wedges there will be eight wedges in total so again this particular wedge will have an area of 12 * s * a in order to get the area of the entire shape we need to multiply it by 8 this will equivalently be 12 * 8 s a but 8s is the perimeter of the Octagon so this simplifies to be 1 12 * the perimeter time the length of the apoem so we have that same formula now let's increase the number of sides until we have a regular Eng gon once again we can divide it up into different wedges the area of this wedge is equal to 12 * s * a for the area of the entire shape we will have 1/2 * the perimeter * a now what happens as n goes to Infinity the perimeter of this polygon will approach the circumference of the circle and eventually will be equal to the circumference of the circle but the circumference of the circle by definition of Pi is equal to 2 pi r furthermore the length of this apothem will get closer and closer and eventually reach the value of the radius so a is going to go to R so substituting those into the formula gives 12 * C * R we substitute in that c is equal to 2 pi r the 1/2 and the two will cancel and the r and r will multiply to become R 2 so we get the area of the circle is equal to Pi * R 2ar wow now here's a method by linard Da Vinci and Sato motion Begin by partitioning the circle into slices you can think about it like slicing up a pizza pie this proof will be similar to Archimedes except you're going to deal with slices of the pizza and not slices of an inscribed polygon now let's rearrange half of the slices and interweave them with the other half of slices we end up with a shape that looks like a parallelogram now what would happen if we increase the number of slices if we interweave the slices we end up with a shape that resembles a parallelogram even further now we can think about this process in the limited case if we have infinitely thin slices and we interweave these slices together of one half of the circle with the other half we're going to end up with a rectangle now this rectangle is exactly made up of slices of the circle so the rectangle has exactly the same area as the circle but the great thing is we know how to calculate the area of a rectangle it will be its length times its width so all we need to do is calculate the dimensions of this rectangle clearly one of the dimensions is the radius of the circle for the other dimension we think about its border being exactly half of the slices of the circle so the other dimension will be half of the circumference the circumference ID 2 is equal to Pi * R so we multiply R by pi * R and that gives the area of the circle is pi r SAR amazing a final method comes from Rabbi Abraham bar hiia hanasi think about the entire area of the circle as being the sum of the areas of concentric Rings now what would happen if we make a cut and unwrap all of these concentric circles into straight line [Music] segments we end up with a triangular like shape now let's imagine that we increase the number of rings and we do this unwrapping process again if we were to take the the limit as this goes to Infinity we would end up with a perfect [Music] triangle and so this triangle has exactly the same area as the circle but what's the area of a triangle it's equal to2 its base time its height the height of this triangle will exactly be the radius of the original Circle and it's base will exactly be the circumference which is 2 pi r so we take 12 * 2 pi r * R and we get the area of the circle is pi r sared wow what's wonderful is this method can actually be made rigorous here's a diagram that I've adapted from a paper about this topic you can actually Define a mapping between the circle and this triangle in a completely rigorous way then you can calculate the Jacobian of this mapping and then it simplifies all to be equal to one finally the area of the circle will be equal to this double integral and this all simplifies to be equal to the area of the triangle and therefore it is in fact true that this method can be made completely rigorous and stand the test of time it is truly amazing that the Ancients were able to figure out the area of the circle is equal to Pi R 2 and it is still a fact that fascinates us to this day thanks for making us one of the best communities on YouTube see you next episode of mind your decisions where we solve The World's problems one video at a time oh

Channel: MindYourDecisions

Views: 65,652

Keywords: mathematics, math, maths, riddle, brain teaser, puzzle, math puzzle

Id: mY_5-KeovKc

Length: 11min 5sec (665 seconds)

Published: Fri Jun 07 2024