The Squared Circle

Discussion in 'Physics & Math' started by Motor Daddy, May 7, 2022.

1. exchemistValued Senior Member

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OK, I've finished my gardening and have an hour to spare before tea, so let's have a look.

By my calculation, the side of the square is approx. 1.777, if I have not made any error applying the Sine Rule and I've done my Tan⁻¹ correctly. (Like any physical scientist, I am alert to the possibility of errors in my algebra and arithmetic.)

Motor Daddy can't do it at all, of course. He just wants to bugger people up and then dance around chortling about it.

I give my working below so others can check it and correct if necessary:-

1) Angle between left hand vertical and 1st rising diagonal is Tan⁻¹ 2.5/2 = 51.34⁰. This is also the angle between the same rising diagonal and left hand edge of the square (alternate angles).

2) Angle between same diagonal and central horizontal is 90-51.34 = 38.66⁰.

3) Triangle formed by left hand edge of square and the 2 diagonals extending to the left edge of the figure is isosceles, with apex angle 2 x 38.66⁰ = 77.32⁰ and base angles 51.34⁰. So we have all the angles. If we can find the length of a side of this triangle, we can find the length of its base by the Sine Rule.

4) Considering now the triangle formed by the left hand corner of the figure, the left hand corner of the square and the first rising diagonal, the upper left angle is 45⁰ and the lower left angle was worked out in (1), so the angle at the corner of the square will be 83.66⁰. So we have all the angles. But we also know the left hand side is 2 units in length. So Sin 83.66⁰/2 = Sin 45⁰/x, where x is the unknown length of the portion of the rising diagonal between the left hand side of the figure and the corner of the square. So x = 2.Sin 45⁰/Sin 83.66⁰ = 1.423.

5) So now, returning to our triangle in (3), we can apply the Sine Rule again and say: Sin 51.34⁰/1.423 = Sin 77.32⁰/s, where s is the unknown length of the base of the triangle formed by the side of the square. So s = 1.423 . Sin 77.32⁰ / Sin 51.34⁰ = 1.777.

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Reported for off topic statements attacking a person and not the idea.

5. James RJust this guy, you know?Staff Member

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You should be thanking exchemist for his help on the problem you posted, Motor Daddy.

You haven't shown us your work - if you've actually done any.

7. SsssssssRegistered Senior Member

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Large square has side length 4 because it fits two circles of radius 1

AC is half the height of the large square = 2
CD is half (width of large square - diameter of circle) = 1
AB is half (height of large square - diameter of circle) = 1
Triangle ACD is similar to ABE so AC/AB = CD/BE
Solve for BE = 0.5
AF is BE + diameter of circle = 2.5

CG = AF = 2.5
FG is half height of large square = 2
HI is half height of red square = s/2
CH is half (width of large square minus width of red square) = 2 - s/2
Triangle CHI is similar to CGF so GF/HI = CG/CH
Sub in and solve for s = 8/4.5 = 1.777... $\neq\sqrt{\pi}$

So circle has not been squared of course and no trig needed to prove it.

Last edited: May 10, 2022
exchemist likes this.
8. exchemistValued Senior Member

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That's much more elegant than my number-crunching.

Nice.

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Square root of 3.14159 = 1.772...
Side length of square = 1.777...

The square is accurate to TWO DECIMAL PLACES! It is 5 THOUSANDTHS of a unit off. It is .005 of a unit off. Break a unit into a thousand equal parts and the side length of the square is only 5 of those parts too long.

You can't even tell me what value pi has unless you TRUNCATE pi...

3.1 > 3.0
3.14 > 3.1
3.141 > 3.14
3.1415 > 3.141
3.14159 > 3.1415

See? For every decimal place you add another number to pi, it gets larger in value.
You can not tell me a FINITE pi value, so you don't even have a value for pi, which means you have no finite area for the circle...But we BOTH know that's BULLSHIT, right? The circle's area is FINITE! If the circle's area is finite that means pi MUST be a finite number! What is that finite number that you claim pi to be in order for the circle to have a finite area????

So according to you the circle is too small too, but it is growing with pi with every decimal place for pi! (rolls eyes)

Last edited: May 10, 2022
10. exchemistValued Senior Member

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Somebody's hovercraft is full of eels.

11. SsssssssRegistered Senior Member

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No mate. 8/4.5=1.78 to 2dp and $\sqrt{\pi}$=1.77 to 2dp so this isn't even accurate to 2dp and the area $(8/4.5)^2$= 3.16 to 2dp is even further out.
You aren't adding digits to $\pi$ you're adding digits to a truncated decimal representation of $\pi$ and you'll never even reach 3.15 by doing that and $(8/4.5)^2\approx 3.16$ is larger even than that.
You're mixing up finite and algebraic. And I read the thread on mathforums that you linked. You can't even understand a repeating decimal so transcendental numbers are beyond you. Go away and buy a high school maths book to go with the high school physics book I suggested you buy in the other thread and when you've learned enough maths to graduate high school then we can talk.

12. Michael 345New year. PRESENT is 72 years oldlValued Senior Member

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How many square units of area is the circle??

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So you're saying the area of a circle is not finite?

15. Michael 345New year. PRESENT is 72 years oldlValued Senior Member

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ONE?????

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Maybe you need to get some learnins...

One divided by three is .333.....

.333... x 3 = .999...

So you CAN'T finish the division of 1 divided by 3. Period.

For you to claim otherwise is total ignorance!

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Nope, try again. The radius is 1.0 unit.

18. Michael 345New year. PRESENT is 72 years oldlValued Senior Member

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If that is what the article says the author of is saying that. I am merely passing on the message

If that is NOT what the article says the author of is saying that. I am still merely passing on the message

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If the circle's radius is 1.0 unit, what is the circle's area?

20. Michael 345New year. PRESENT is 72 years oldlValued Senior Member

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Since the (a) circle is a single unit it has one unit of area

21. SsssssssRegistered Senior Member

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$\pi$. Duh.
Like I say you don't know the difference between algebraic and finite.
Like I say you need to learn high school maths. It's easy enough to prove 3 times the infinitely repeating decimal 0.33333.... is 1 and it was proved for you in the mathforums thread. You either didn't understand it or are pretending you don't understand it because you're either as dumb as a bag of spanners or dumb enough to get a kick out of pretending to be as dumb as a bag of spanners which is even dumber.

exchemist likes this.

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You calculate the circle's area using the formula Area=pi(r^2)

The radius is 1.0 unit, so how many square units of area is the circle?

Last edited: May 10, 2022