## Proof of calculating area and volume

If r is the radius of a circle,

For circle perimeter we use; ''2*pi*r''

For circle area we use; ''pi*r²''

And for sphere we use; (4/3)*pi*r³

but how do we calculate them?
 Do you mean, how did we find these formulas? If YES, then, they arise from integration
 Blog Entries: 9 Recognitions: Homework Help Science Advisor Yes,by integration. Daniel. P.S.The circle does not have an area...

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## Proof of calculating area and volume

 Quote by dextercioby P.S.The circle does not have an area...
What !?

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 Quote by sitokinin If r is the radius of a circle, For circle perimeter we use; ''2*pi*r''
The circumference of the circle is very easy, because it follows from the definition of $\pi$.

 For circle area we use; ''pi*r²''
This actually needs calculus to be rigorous but there exists a cute "proof" that I learned in the local equivalent of elementary school. Take equal miniscule (at the limit, infinitesimal) segments of the circle. Each of these is shaped like a little slice of pizza. When you place two "slices" in opposite orientations, aligned along the radius, you will construct something that looks like a parallelogram. As you take more and more slices, it will approach a rectangle at the limit. The rectangle will have smaller side equal to r and larger side equal to $\pi r$, giving the area of $\pi r^2$. Note that while this helps you to "see" the proof, it really uses basic concepts of calculus explained simply. To do it properly really does require calculus.

 And for sphere we use; (4/3)*pi*r³
There is an elegant proof due to the Greeks that does not involve calculus. Read about it here : http://mathcentral.uregina.ca/QQ/dat...01/rahul1.html

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 Quote by Curious3141 What !?
1.What is the definition of a circle??

Daniel.

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 Quote by dextercioby 1.What is the definition of a circle?? Daniel.
I have absolutely no idea what you're on about, but even Mathworld calls it the "area of the circle". http://mathworld.wolfram.com/Circle.html
 Blog Entries: 9 Recognitions: Homework Help Science Advisor OMG,they can't make the difference between a circle and disk... THE CIRCLE IS A CURVE,A UNIDIMENSIONAL OBJECT.CAN U CONSIDER THE AREA OF A CURVE???CAN U CONSIDER THE LENGTH OF A SURFACE??CAN U CONSIDER THE VOLUME OF A SURFACE?CAN U CONSIDER THE VOLUME OF A CURVE??? ME NEITHER. So how can someone speak about the area of a circle?? Daniel. PS.What about the triangle??The square??The polygons??Do they have area??
 Recognitions: Homework Help Enclosed area. No need for the outraged splitting of hairs.
 Blog Entries: 9 Recognitions: Homework Help Science Advisor AAAAAAAAAAAAAAAAAAAAaaaaaaaaaaaaaaaaaaaaaahhhhhhhh,well that's something else... I'm glad you agree. Daniel.
 Recognitions: Homework Help Good. Now what about the binomial thing below ?
 Blog Entries: 9 Recognitions: Homework Help Science Advisor Go & check it out... Daniel.
 Recognitions: Homework Help Perfect.
 Recognitions: Gold Member Science Advisor Staff Emeritus If you really want to split hairs, there's an important difference between having zero area and not having area. (The circle has zero area)
 hey, so, how do I call the enclosed area of a triangle, square, rectangle ...etc
 Blog Entries: 9 Recognitions: Homework Help Science Advisor Please,Hurkyl,show us that the circle has zero area using the double integral construction with Riemann sums... Daniel.

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