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# Dividing the Area of a Circle watch

1. How would I divide the area of a circle in the ratio 3:1 with one straight cut (the smaller piece will have an area equal to a quarter of the area of the circle)?

I am trying to find the distance from the edge of the circle to the cut which I have called . (See attached for my diagram).

My plan was to work out the area of the sector OPQ and then subtract the area of the triangle. By equating this to I would then get the length .

I have worked out the area of the triangle OPQ using Pythagoras and to be:

For the area of the sector OPQ I used:

Therefore:

I tried rearranging for but didn't really get anywhere. Also I don't know so was never going to reach an answer for anyway.

EDIT: P and Q are meant to be where the cut meets the circle and is meant to be the angle at O.
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2. (Original post by SherlockHolmes)
How would I divide the area of a circle in the ratio 3:1 with one straight cut (the smaller piece will have an area equal to a quarter of the area of the circle)?

I am trying to find the distance from the edge of the circle to the cut which I have called . (See attached for my diagram).

My plan was to work out the area of the sector OPQ and then subtract the area of the triangle. By equating this to I would then get the length .

I have worked out the area of the triangle OPQ using Pythagoras and to be:

For the area of the sector OPQ I used:

Therefore:

I tried rearranging for but didn't really get anywhere. Also I don't know so was never going to reach an answer for anyway.

EDIT: P and Q are meant to be where the cut meets the circle and is meant to be the angle at O.
Without loss of generality you can take the radius =1. Then use
for the area of the triangle.
The resulting equation in theta could then be solved numerically by various methods.
3. (Original post by brianeverit)
Without loss of generality you can take the radius =1. Then use
for the area of the triangle.
The resulting equation in theta could then be solved numerically by various methods.
Thanks for the reply. I have:

How would this be solved?
4. (Original post by SherlockHolmes)
Thanks for the reply. I have:

How would this be solved?
Needs to be solved numerically rather than algebraically
5. Is this a level or degree??
6. (Original post by TenOfThem)
Needs to be solved numerically rather than algebraically
I have used the Newton-Raphson Method and got an answer of 2.31... radians. Then using some trig I have got to be 0.586... which seems to be about right.

Thanks for the help all.
7. (Original post by benwalters1996)
Is this a level or degree??
It's probably classed under A level knowledge. It was a question given by my teacher.
8. (Original post by SherlockHolmes)
I have used the Newton-Raphson Method and got an answer of 2.31... radians. Then using some trig I have got to be 0.586... which seems to be about right.

Thanks for the help all.
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