Proof of calculating area and volume

sitokinin

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?

Popey

Do you mean, how did we find these formulas?

If YES, then, they arise from integration

dextercioby

Yes,by integration.

Daniel.

P.S.The circle does not have an area...

Curious3141

What !?

Curious3141

The circumference of the circle is very easy, because it follows from the definition of [itex]\pi[/itex].

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 [itex]\pi r[/itex], giving the area of [itex]\pi r^2[/itex]. 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.


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

dextercioby

1.What is the definition of a circle??

Daniel.

Curious3141

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

dextercioby

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??

Curious3141

Enclosed area. No need for the outraged splitting of hairs.

dextercioby

AAAAAAAAAAAAAAAAAAAAaaaaaaaaaaaaaaaaaaaaaahhhhhhhh,well that's something else...

I'm glad you agree.

Daniel.

Curious3141

Good. Now what about the binomial thing below ?

dextercioby

Go & check it out...

Daniel.

Curious3141

Perfect.

Hurkyl

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)

vincentchan

hey, so, how do I call the enclosed area of a triangle, square, rectangle ...etc

dextercioby

Please,Hurkyl,show us that the circle has zero area using the double integral construction with Riemann sums...

Daniel.

dextercioby

The "enclosed area of a triangle/square/rectangle" ???

The "enclosed volume of a cone/cylinder"??The volume of a ball??

Daniel.

PS.I liked the part with the physicists... ball