Finding area within a circle from a point NOT at the centre

Coach Sanders

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Hey, I've been working on this question for years and have been unable to find anything regarding area within a circle from a point on a chord.

It involves cutting a pizza into 5 by starting with a segment equal to one fifth the area of the pizza (this is estimated since I haven't solved this problem). Then, from the midpoint of that chord, cut the remaining four fifths in half, and those 2 in half again (from that same point). The result needs to be 5 equal pieces.

The question then is: what is the height of the segment? and the measure of the angles dividing the other 4 pieces?
 

Dr.Peterson

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Do you mean something like this?

FMH117842.png

I've worked out approximate values for CD and the angles shown; they can't be calculated exactly, but require numerical methods (I used Desmos).

Now, what help do you want from us? The main formula I used was for the area of a circular segment (I used equation 14). I also used formulas for area of a sector and of a triangle.
 

Coach Sanders

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Thank you,
This is exactly what I mean, so I'm wondering how you come up with those measurements. If they can't be measured exactly, why is that? The greatest thing about math, is that there should always be an answer; a method. If a computer can generate an image like that, where all 5 pieces have the same area, shouldn't there be a way to get there?
 

Dr.Peterson

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Thank you,
This is exactly what I mean, so I'm wondering how you come up with those measurements. If they can't be measured exactly, why is that? The greatest thing about math, is that there should always be an answer; a method. If a computer can generate an image like that, where all 5 pieces have the same area, shouldn't there be a way to get there?
You can get there, but not exactly, and not by a formula. Computers are great at approximating things!

I wrote a couple equations, as I described, and then used a computer program to find approximate solutions. It is a dirty little secret that algebra can't solve every equation you can write; in fact, the only equations we usually show students are those that we know how to solve. The equations for the height of the first segment, and for the angles in the others, are transcendental equations, which require numerical approximation methods.

I could also have constructed the picture (in GeoGebra) by just placing points and moving them around until I got the right areas; I chose instead to get numerical values first. I can't show you the equations I used at the moment, because I'm not at home where I did them.
 

Coach Sanders

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Thank you very much Dr. Peterson. It actually makes me feel better that, despite many years of trying to solve what seemed to be a simple math problem, this is a problem "Math" has not tried to figure out. Aside from a desire to get 5 equal pieces of pizza (which I will generally eat by myself anyway), it must not have any other useful applications.

I will sleep better now.
 

Dr.Peterson

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Just to be complete, here are the two equations I used, where I'm taking the radius as 1, d is the distance from the center to my point D, C is my angle at point C, and D is my angle at D:

\(\displaystyle \cos^{-1}(d) - d\sqrt{1 - d^2} = \pi/5\)​
\(\displaystyle C + d\sin(C) = 2\pi/5\)​
\(\displaystyle D = \tan^{-1}\frac{\sin(C)}{d + \cos(C)}\)​
The approximate solution is d = 0.492, C = 0.878 radians = 50.3 degrees, D = 34.23 degrees.
 

Coach Sanders

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Thank you Dr. Peterson.

I just watched an amazing video on the 3Brown1Blue channel regarding the Basel Problem, and how Pi fits in there, and another video on the relationship between circles and ellipses.

I need to work on this some more, as I am now wondering if point D was an eccentric point, would the other calculations be easier to make, and with greater accuracy, but I wanted to share this eureka moment with you before I proceed.

I may sleep better now, but I am excited to be awaken by a new approach.
 
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