Area of a Circle - The Pizza Method
The formula for the area of a circle is easy to derive if you use the "pizza method".
A circle is a famous shape, because the ratio of its two most obvious features, the circumference and the diameter, is the same for all circles and is the famous number pi.
pi=circumference/diameter=3.14159265359...
The decimal places go on forever.
So, given the diameter of a circle we can easily calculate its circumference.
circumference=pi*diameter=pi*2*radius
But what about the area? We can get this by the following clever method..
Just imagine the circle of radius r is a pizza and we cut it into an even number of slices, 2*n where n is an integer. Now take two slices and place them on the table, but position them in opposite directions so they make an approximate rectangle. The length of this rectangle is r and its width is equal to the width of a slice. Of course it's not a perfect rectangle because the top and bottom edges are curvey, but to a good approximation we can say..
area of one rectangle=r*(width of a slice)
But approximately..
width of a slice=circumference/(2*n)=(2*pi*r)/(2*n)=pi*r/n
So, area of one rectangle=r*(width of a slice)=pi*r^2/n
Now, as we increase n the pizza gets cut into thinner and thinner slices and these approximations get better and better. In fact, when n becomes a truly massive number we can call these approximation exact, and so..
area of all the rectangles=n*(area of one rectangle)=pi*r^2
This area must be the same as the circle, so we have..
area of circle=pi*r^2
Which is the correct formula for the area of a circle.
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Content written and posted by Ken Abbott abbottsystems@gmail.com
Internet Marketing Consultant