Pi r^2 /3

  • Thread starter Organic
  • Start date
  • Tags
    Pi
In summary, the conversation discusses the relationship between a circle and its sub-area, which can be found by dividing the circle's area by 3. The conversation also explores the concept of natural numbers and their connection to polygon areas. It asks about the potential usefulness of a mathematical constant and its relation to the CH problem. The conversation also discusses the Archimedian spiral and its area, which is shown to be 1/3 of the area of the circle.
  • #1
Organic
1,224
0
Please look at the attached pdf.

You will find in it a circle and some sub-area inside it.

The sub-area exists between the radius and a curve.

The curve is connected to both sides of some radius, and goes through the intersection points that exists between n radii and n-1 inner circles, where each radius is divided by the inner circles to n equal parts.

I have found that the sub-area(magenta) = circle's-area/3(cyan)

1) Can someone show why the magenta area = 1/3 of the cyan area ?

2) We can take any number of inner radii-circles intersection points, and create some border, which is made of straight lines between these points.

By doing this, we get a closed polygon (an area).

Now we take some closed polygon, find the total number of its vertexes and omit 2 (tolal - 2 = n).

By doing this, we get some Natural number n which is conncted to some polygon's area S (please see the attached pdf in the next post, called natural-areas.pdf.pdf).

Through this way we can put in 1-1 correspondence some n with some S.

When have this map, we can ask:

S1 is the area of some polygon, where the number of totel-2=aleph0.

S2 is the area of some polygon, where the number of totel-2=2^aleph0.


3) Is S1 = S2 ?

4) If the answer to (3) is no, then what is the difference between the two magenta areas, and how this difference related to the CH problem ?

5) Do you think that we have here some useful mathematical constant ?


Thank you.


Organic
 

Attachments

  • 123.pdf.pdf
    4.1 KB · Views: 354
Last edited:
Physics news on Phys.org
  • #2
Here you can find a pdf file, which shows the connection between some natural number to some area.
 

Attachments

  • natural-areas.pdf.pdf
    2.9 KB · Views: 336
Last edited:
  • #3
The spiral you have is the "Archimedian" spiral:
r= (R/2 &pi)&theta (R is the radius of the large circle).

It is fairly easy to show that the area is, in fact, (2/3)&pi R2, 1/3 the area of the circle.
 
  • #4
Hi HallsofIvy,

Thank you for your reply.

Can you please answer to 3-5 questions ?

Thank you.


Organic
 
Last edited:

1. What is the formula for "Pi r^2 /3"?

The formula for "Pi r^2 /3" is the formula for the volume of a cone. It is calculated by multiplying the square of the radius by pi and then dividing by 3.

2. How is "Pi r^2 /3" related to the area of a circle?

"Pi r^2 /3" is not directly related to the area of a circle. However, if you multiply the formula by the height of the cone, you will get the formula for the volume of a cylinder, which is related to the area of a circle.

3. Why is "Pi" used in the formula for "Pi r^2 /3"?

"Pi" is used in the formula because it represents the ratio of a circle's circumference to its diameter. It is a constant value that is approximately equal to 3.14159.

4. Can "Pi r^2 /3" be used to calculate the volume of any cone?

Yes, "Pi r^2 /3" can be used to calculate the volume of any cone, as long as the measurements of the radius and height are known.

5. What is the unit of measurement for "Pi r^2 /3"?

The unit of measurement for "Pi r^2 /3" depends on the units used for the radius and height. The resulting volume will be in cubic units, such as cubic inches or cubic centimeters.

Similar threads

Replies
7
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
911
  • Linear and Abstract Algebra
Replies
1
Views
862
  • Sci-Fi Writing and World Building
Replies
3
Views
667
  • Precalculus Mathematics Homework Help
Replies
4
Views
1K
  • Special and General Relativity
Replies
29
Views
1K
Replies
3
Views
214
  • Set Theory, Logic, Probability, Statistics
Replies
15
Views
1K
Replies
0
Views
515
Replies
4
Views
2K
Back
Top