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Approximating the Area of a Circle using Rectangles
Adjust the number of rectangles used to approximate the area of the circle
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Main Concept
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The area of a circle can be approximated by rectangles. As the number of rectangles approaches infinity the total area of all the rectangles approaches the actual area of the circle.
Using integration, the exact area of the circle can be found. The exact area of a circle is  , where r is the radius.
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Proof
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Let r be the radius of the circle, and let n be the number of approximating rectangles.
The height h of each rectangle can be defined as:
The length ![l[k]](/support/helpjp/helpview.aspx?si=8362/file02223/math92.png) of the kth rectangle located at height ![y[k]](/support/helpjp/helpview.aspx?si=8362/file02223/math96.png) can be found from
![((1/2)*l[k])^2 = r^2-y[k]^2](/support/helpjp/helpview.aspx?si=8362/file02223/math102.png)
The area of the kth rectangle is:
![A[k] = l[k]*h and l[k]*h = 2*sqrt(r^2-y[k]^2)*h](/support/helpjp/helpview.aspx?si=8362/file02223/math119.png)
Therefore the total area of n rectangles is:
![Area = 2*(sum(sqrt(r^2-y[k]^2)*h, k = 1 .. n))](/support/helpjp/helpview.aspx?si=8362/file02223/math142.png)
In the limit as  , we get the area of the circle:
![A[circle] =2 (∫)[-r]^(r)sqrt(r^(2)-y^(2)) dy](/support/helpjp/helpview.aspx?si=8362/file02223/math154.png)
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Number of Rectangles =
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