20060928, 21:51  #23 
6809 > 6502
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Aug 2003
101×103 Posts
2226_{16} Posts 
Common usage, area of a circle refers to the area of the disk.
At: http://en.wikipedia.org/wiki/Circle they aknowledge both usages. Food for thought, like the concentric circles, is it possible to construct polygons of the appropriate size? Or a half size circle that is bisected by a diameter? 
20060929, 02:08  #24 
Jun 2003
The Texas Hill Country
2×541 Posts 
It is trivial to construct concentric circles of radius 1, sqrt(2), and 2.
Further, any diameter can be used to "halve" any of the circles. But, if you use this type of construct, is not the proposed solution one with "radial" lines. (Any part of a diameter) 
20060929, 02:15  #25  
Jun 2003
The Texas Hill Country
2×541 Posts 
Quote:
I think that that implies that it cannot be done with any other polygon, either. 

20060929, 08:35  #26 
Bamboozled!
"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across
2×61×83 Posts 
Unfortunately, you did not say "area of a circle" or "interior of a circle" or any other such phrase. You asked for the circle to be divided and it is that problem which I addressed.
Mathematics requires pedantic precision. A good case can be made that mathematics is pedantry carried to its extreme. Paul 
20060929, 12:53  #27  
Nov 2003
2^{6}×113 Posts 
Quote:
The preliminary U.S. Olympiad team exam had multiple choice answers. One question asked: What is the maximum number of regions into which a circle may be cut by n lines? The correct answer was not among those given. The reason is that the one who made up the question intended "disk" instead of circle. I left this question blank, and attached a note to the exam explaining why. The note included a solution for the question *AS ASKED*. The person grading the exam would not give me credit, because it was clear to him that the question intended "disk". I argued that in mathematics if one meant disk, one said "disk" and not "circle". The teacher grading the exam fired back that there was *no difference* and I then fired back that the teacher had no business teaching math because he was incompetent. This led to a discussion with the department chair, my math teacher, the high school principal, and my parents..... The department chair gave me full credit for my answer and actually told the teacher grading the exam that he should have listened to me; that a disk is not a circle. He also added that the teacher should have listened given who I was. (I had quite a reputation in the school math & science depts). The teacher fired back that I was "just another student, he is the teacher and was not to be argued with". At that point, *my* teacher supported me. He said that that I was not "just another student", that I was then taking my 3rd year of calculus and it was virtually certain that I knew more math than the grader and that he should have listened to me given who I was. Apparently the grader was a new teacher who didn't know me from a hole in the ground. The principal opined that what I said to the grader was rude. I agreed, but said that I was just responding to the rudeness of the teacher in not being willing to listen. I said that I would apologize if the teacher apologized. The teacher refused, taking the position "I don't have to apologize to a student". At that point I said that a teacher who was unwilling to listen to a student should not be teaching....The teacher responded that he did listen, decided that I was wrong (my math teacher and the department chair rolled their eyes a bit) and that should have ended the discussion. That teacher was not teaching the following year... I later had a private talk with my teacher and he agreed that the grader should not be teaching given his attitude towards students and his failure to recognize an elementary mathematical fact. 

20060929, 16:16  #28  
∂^{2}ω=0
Sep 2002
República de California
2·59·83 Posts 
Quote:
Quote:


20061002, 17:55  #29 
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}·3^{3}·19 Posts 
4 not so easy pieces?
Off hand I would say that this problem can be solved by using the principle of Hippocrates' lunes  a straight line, which is not radial, and a curve. Mally 
20061003, 15:45  #30  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
Hippocrates' lunes.
Quote:
Splendid problem, Uncwilly, even an archaic one, solved in the time of Aristotle. Construction : Draw a semicircle of the given circle and inscribe an isosceles triangle on the diameter. On any of its equal sides construct a semi circle. This semi circle will be equal to the quadrant of the 1st semi circle as required. Sorry I dont have the expertise to draw it out. This problem is based on the proof that the crescent ( lune) formed is equal to half the isosceles triangle constructed. Mally 

20061003, 21:28  #31 
6809 > 6502
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Aug 2003
101×103 Posts
2226_{16} Posts 
Attached is the image that I stumbled upon whilst looking for around the inet.
The solution look simple to construct and has no straight lines. While it has X or rotational symetry, it is not radial (star formed). 
20061003, 22:32  #32  
Nov 2003
2^{6}·113 Posts 
Quote:
As I said earlier, 5 pieces is easier. 

20061003, 22:52  #33 
Jun 2003
The Texas Hill Country
2·541 Posts 
That is left as an exercise for the pupil .... :)
Actually, it is rather easy. First construct the diameter. Then divide it into 8 equal parts. That gives you all of the centers of the various circular segments. 
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