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 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
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Quote:
 Originally Posted by Country Boy (But the larger n is the smaller those differences are.
Uh-oh. Proto-calculus again! It's amazing that these ideas are well over 2000 years old yet only got completely formalized in the late 19th century.

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