# 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)