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bison

Math Challenge-- Pi R Round

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We're al familiar with the formula for determining the area of a circle: A = Pi * r 2. Can anyone give a alternate formula for the area of a circle that does not mention its radius, diameter or circumference? clue: It is possible to do this, given enough time.

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Given a circle with center 0,0, and any given point x,y on the circle, the area would be (x2+y2)xPi. That's my hubbies guess :)

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Slack jawed stare... :huh:

I could have done this in college... no, really!! Really!!! Would've took awhile but I could've.

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Divide the circle using an equilateral triangle (a circumcircle). Each side of the triangle (which is not a radius) has a length L.

To measure the area of the circle use the formula (L / 3-2)2 * pi.

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Thank you to everyone who responded to this thread. Besides providing a problem for solution, I wanted to see if anyone would hit upon a particular way of deriving the area of a circle I was thinking of. No one did.

It's interesting that Leonardo mentions a circumscribed equilateral triangle, but uses it in a different way than I had in mind.

If you draw the largest possible square inside the space bounded by the triangle and the side of the circle, the circle will have Pi times the area of the square. One can translate this from graphics to numbers, by measuring the side of the square, multiplying this figure by itself, and multiplying that product by Pi. The result is the area of the circle.

It's interesting how this diagram was first presented. We do not know who to credit for it, because it was not published, nor put on the internet with anyone's name attached to it. It simply appeared one summer's day in 1998 pressed into a field of grain in Old Sarum, Wiltshire, in the U.K..

One may think what they will about 'crop circles'. Consider this, though: We have the fact of a solution for the area of a circle that no one on this planet had apparently managed to think of in all the long history of mathematics. We also have the fact of its appearance by a means that some, at least, have suspected of being connected to an off-world intelligence.

Edited by bison

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This thread is very interesting, but...

We have the fact of a solution for the area of a circle that no one on this planet had apparently managed to think of in all the long history of mathematics.

I think that statement is a little bit to strong.

This field of mathematics (in human history, anyway) is concerned (for historical reasons) with constructing shapes with compass and straight-edge.

It is not very difficult to draw a square whose area is 1/pi times the area of a circle. You can construct line that is the diameter of that circle, bisect the line, and construct a square based on one of the halves of that line. The area of the square will be r2 where r is the radius of the circle.

The field of mathematics is much more interested in deducing the area of a circle without explicitly using pi.

Pi is a fascinating number, and rather hard to calculate without using calculus (and, preferably, a fast computer). It is evident after even a cursory look at the history of mathematics, that many societies, from antiquity right up until the 1500s, knew exactly what pi was, and why it was important, but did not have a value for pi more accurate than 3 or 4 decimal places.

That is why there was such a prolonged search for a method of constructing a polygon whose area was exactly a rational multiple of the area of a given circle's.

The method you present here is just ``playing with shapes''. I don't doubt that many mathematicians (and school children for that matter) stumbled upon this method, but it doesn't really do anything - and as I pointed out above, it is a rather long way of doing a relatively simple thing.

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I think that the statement that no one has thought of this approach to finding the area of a circle, qualified by the word apparently, is about as strong as it needs to be. I did considerable research, and found no evidence that anyone was aware of it. If it is such a commonplace, easily discerned thing, why should the least mention of it be so elusive? I put the question in my original post to people on several websites. A number of complex and ingenious solutions were proposed, but not this relatively simple one.

The diagram does contain the Pi ratio, graphically represented as the ratio of the area of the circle to that of the square. Defining Pi as a numerical string does not appear to be its purpose.

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If it is such a commonplace, easily discerned thing, why should the least mention of it be so elusive? I put the question in my original post to people on several websites. A number of complex and ingenious solutions were proposed, but not this relatively simple one.

The method I described is simpler than the one you did, so why did you not think of that method?

What you are suggesting is that everyone should 'know' every answer to a question, which is patently absurd. And the way you asked for a solution suggested that solution was complex, so it is natural people would search for complex answers and perhaps not see the obvious.

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I didn't mean to suggest that everyone should know the answer I was looking for. It does seem that some previous knowledge of it was a natural thing to expect, especially if, as Sepulchrave suggested, it was something so obvious that many, including students have happened upon it.

Besides putting my question on several discussion boards, I did a good deal of other searching, without finding any evidence that this method for finding the area of a circle was known.

I'm sorry if I seemed to be asking for a more complex answer than was wanted. It was not my intention to mislead anyone. It is still not clear to me what I wrote that would have given that impression.

Edited by bison

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I didn't mean to suggest that everyone should know the answer I was looking for. It does seem that some previous knowledge of it was a natural thing to expect, especially if, as Sepulchrave suggested, it was something so obvious that many, including students have happened upon it.

Besides putting my question on several discussion boards, I did a good deal of other searching, without finding any evidence that this method for finding the area of a circle was known.

I'm sorry if I seemed to be asking for a more complex answer than was wanted. It was not my intention to mislead anyone. It is still not clear to me what I wrote that would have given that impression.

Well, saying "given enough time" suggests finding the solution is time-consuming - and most people will equate this with complex in the context of mathematics. It might not have been your intention to convey this impression, but I feel you did.

And I don't mean to suggest you misled anyone, just that what you expected was not what you might have conveyed to others.

Edited by Leonardo

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