April 10th, 2018, 02:41 PM  #1 
Global Moderator Joined: May 2007 Posts: 6,683 Thanks: 658  area of circle
Can the area of a circle $\pi R^2$ be derived from the definition circumference=$2\pi R$, without calculus?

April 10th, 2018, 05:55 PM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 896 
Imagine cutting the disk into n, a large integer, "slices of pie" then put them back together alternating "point down" with "point up" so you get something like a rectangle (Except that the sides slant and the top and bottom are "rippled". But the larger n is the smaller those differences are.) The height of the rectangle is the radius, r. The top and bottom together make up the circumference, $2\pi r$, so each is $\pi r$. The area of that "rectangle" is $r(\pi r)= \pi r^2$. In the limit, as n goes to 0 we get the area of the disk, $\pi r^2$. 
April 10th, 2018, 06:03 PM  #3 
Senior Member Joined: Aug 2012 Posts: 2,156 Thanks: 630  

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