Proof of calculating area and volume

1. Jan 16, 2005

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?

2. Jan 16, 2005

Popey

Do you mean, how did we find these formulas?

If YES, then, they arise from integration

3. Jan 16, 2005

dextercioby

Yes,by integration.

Daniel.

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

4. Jan 16, 2005

Curious3141

What !? :yuck:

5. Jan 16, 2005

Curious3141

The circumference of the circle is very easy, because it follows from the definition of $\pi$.

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.

There is an elegant proof due to the Greeks that does not involve calculus. Read about it here : http://mathcentral.uregina.ca/QQ/database/QQ.09.01/rahul1.html

6. Jan 16, 2005

dextercioby

1.What is the definition of a circle??

Daniel.

7. Jan 16, 2005

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

8. Jan 16, 2005

dextercioby

OMG,they can't make the difference between a circle and disk... :yuck:

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??

:yuck: :yuck: :yuck: :yuck:

Daniel.

PS.What about the triangle??The square??The polygons??Do they have area??

9. Jan 17, 2005

Curious3141

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

10. Jan 17, 2005

dextercioby

AAAAAAAAAAAAAAAAAAAAaaaaaaaaaaaaaaaaaaaaaahhhhhhhh,well that's something else...

Daniel.

11. Jan 17, 2005

Curious3141

Good. Now what about the binomial thing below ?

12. Jan 17, 2005

dextercioby

Go & check it out...

Daniel.

13. Jan 17, 2005

Curious3141

Perfect.

14. Jan 17, 2005

Hurkyl

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)

15. Jan 17, 2005

vincentchan

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

16. Jan 17, 2005

dextercioby

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

Daniel.

17. Jan 17, 2005

dextercioby

The "enclosed area of a triangle/square/rectangle" ??? :tongue2:

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

Daniel.

PS.I liked the part with the physicists... :tongue2: ball

18. Jan 17, 2005

Hurkyl

Staff Emeritus
It reduces to showing that the outer area of the circle is zero. Recalling the definition, the outer area of a set X is defined as follows:

Consider any grid G of rectangles that covers X. Let A(G, X) be the total area of the rectangles of G that have a nonempty intersection with X. Then, the outer area of X is defined to be the infimum of A(G, X) over all grids G.

Then, you just need to show that, as you refine the grid, the number of such rectangles grows roughly linearly, while their areas decrease quadratically.

19. Jan 17, 2005

vincentchan

don't avoid my question... how people called the area enclosed by a triangle? I searched through google and return nothing, also.. is triangle an angle or a curve? if I have a closed shape contains 3 angle and 3 curve(not straight) on a flat plane, do we call it triangle? what if the shape is not closed? Could someone tell me what is a triangle... what is its definition?

20. Jan 17, 2005

Hurkyl

Staff Emeritus
Usually, one specifies precisely what they mean by "rectangle" (and similar words) before using it. (Like I should have done in my previous post)

21. Jan 17, 2005

gerben

22. Jan 17, 2005

dextercioby

I didn't...

The enclosed area of a triangle???

That's because it's not something specific.Any closed curve encloses a domain of (hyper)surface.Any (hyper)surface has an area and any domain (open/closed) has an area.Ergo,we should call it "area of the surface enclosed by a triangle",but that's too long and we say "area enclosed by a triangle"...

It is a closed curve which has three tops/peaks/summits (which is the word?? ),three angles and three sides...

Nope.I think it has to do with the fact that its sides must be geodesic curves on the (hyper)surface in which the triangle lies...

Nope.It wouldn't be a triangle.

Maybe my/this post gave you a (hopefully correct) idea.

Daniel.

Last edited: Jan 17, 2005
23. Jan 17, 2005

vincentchan

dex, you are rock...I like you point out the concept of "geodesic curves", yeah, this solved my straight line problem in the triangle... howabout angles... how do you define angle?

could i define it like this: an angle is a continue curve which is not smooth...

24. Jan 17, 2005

dextercioby

I'm not looking for a textbook definition.I'll let u pick it up.Consider a (hyper)surface and the bunch of curves on it.These curves intersect each other.IIRC the definition of the angle between two curves on a (hyper)surface is the angle betweent the two tangent LINES (THESE MUST BE LINES) at the two curves in the intersection point.Note the fact that the concept of geodesic curves plays no role.Think about the 2D sphere (topological definition).The curves on this surface are from very simple (the geodesics,namely the big circles) to very complicated.The idea that the the angle between two of these curves is not given in terms of geodesics,but in terms of lines has deeper roots and allows the definition of angle between geodesic curves as well.
The roots are found in the fact that the lines i'm talking about care actually vectors in the tangent space of the manifold (2 sphere) in that point of intersection.
In the simple 2D plane u can have angles between geodesics (lines) and general curves (from simple to very complicated) given in terms of the lines tangent of the curves in the intersection point.The tangent space of the 2D plane coincides to the plane itself.
However,for a triangle,the sides must be geodesics and the angles in the triangle are defined in terms of the angle between the geodesics.Think of the astroid,can u think of it as being a "curved" quadrilater??Compute its angles.They are all zero...The sum would not be 360° as it should.Yet it would be in a plane,a closed plane curve...

Geometry is beautiful.

Daniel.

25. Jan 17, 2005

sitokinin

Thanks for the short explanation.

PS: I know the difference between the full circle and the empty circle, but I looked at the dictionary it has just one meaning for both!

Meanwhile when we calculate the area of a triangular, do we add the area of the line segments? I mean does the total area include the inside of the triangular and line segments or just the inside of the triangular?