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April 10th, 2018, 02:41 PM   #1
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area of circle

Can the area of a circle $\pi R^2$ be derived from the definition circumference=$2\pi R$, without calculus?
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April 10th, 2018, 05:55 PM   #2
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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, $\pi r^2$.
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April 10th, 2018, 06:03 PM   #3
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Originally Posted by Country Boy View Post
(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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area, circle

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