3 page proof of the Circle area for amateurs

[Xannari Ferrows]

I bet I can prove the formula for the area (region) of a circle in 3 imgur pages. I know it's not necessary, but I also bet you've never questioned it yourself. Where does it come from? How was it formulated? Is it even true? Have we been lied to? Well... no. And here's why:

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Note: I made this in like 10 minutes, so hence the name "quick math."

Edited February 21, 2015 by Xannari Ferrows

[Tommygun]

.......... Where does it come from? How was it formulated? Is it even true? Have we been lied to? Well...

I blame the Greeks for these conspiracies.............Beware of Greeks bearing circles!

[Red Iron Crown]

I have no idea how those diagrams prove anything. Can you explain a little what's going on in them?

[Xannari Ferrows]

I have no idea how those diagrams prove anything. Can you explain a little what's going on in them?

Certainly^^ They have been updated with a basic explanation, but if it's still confusing, you can ask about anything you want.

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I blame the Greeks for these conspiracies.............Beware of Greeks bearing circles!

And my day has already been made...

[Ralathon]

I have no idea how those diagrams prove anything. Can you explain a little what's going on in them?

He's basically doing this:

[Xannari Ferrows]

He's basically doing this:

There's a legit video on this? Man, I need to get out more... Also, this guy looks cool. I need to watch him...

[PB666]

I bet I can prove the formula for the area (region) of a circle in 3 imgur pages. I know it's not necessary, but I also bet you've never questioned it yourself. Where does it come from? How was it formulated? Is it even true? Have we been lied to? Well... no. And here's why:

http://imgur.com/a/xYT4y

Note: I made this in like 10 minutes, so hence the name "quick math."

I can do it to the resolution of computer in 16 to 32 excel lines (6 cells wide).

I can derive any sine or cosine the same with you using the pi or trig functions built in to the computer

Here's a simpler problem, can you solve the gravitational constant to 6 digits +/- 7 digit resolution

[Xannari Ferrows]

Here's a simpler problem, can you solve the gravitational constant to 6 digits +/- 7 digit resolution

I don't really know what this question is asking. If it was a question, or just worded wrong, I don't know. If it means what I think it does, then your answer is 0.06738 x 10^-9. Then again... I'm not even sure if this is a question...

[Red Iron Crown]

Where does the 26.16 inches come from? The radius is unspecified.

[Xannari Ferrows]

Where does the 26.16 inches come from? The radius is unspecified.

Are you a ninja or something?...

The second page give 26.16 inches as our Circumference of the light blue line, as specified in the texts. It is emphasized by the red layer.

Your radius is irrelevant. You only need to know that it is equal to the height of your triangle.

[GoSlash27]

Unless I'm mistaken, this proof is circular logic. It assumes at the outset that a=pi*r^2 by setting the circumference to 26.16.

Best,

-Slashy

[Red Iron Crown]

Are you a ninja or something?...

The second page give 26.16 inches as our Circumference of the light blue line, as specified in the texts. It is emphasized by the red layer.

Your radius is irrelevant. You only need to know that it is equal to the height of your triangle.

You equate 26.16 to 2*pi*r later in your proof, the radius is very much relevant. I don't see how the 26.16 is relevant to the proof, you should just leave circumference as 2*pi*r, IMO.

Unless I'm mistaken, this proof is circular logic. It assumes at the outset that a=pi*r^2 by setting the circumference to 26.16.

I don't think so, he uses c=2*pi*r but not a=pi*r^2.

[GoSlash27]

You equate 26.16 to 2*pi*r later in your proof, the radius is very much relevant. I don't see how the 26.16 is relevant to the proof, you should just leave circumference as 2*pi*r, IMO.

I don't think so, he uses c=2*pi*r but not a=pi*r^2.

I see what you mean. The 26.16 is completely superfluous.

What matters is that it is assumed that

1) the area of a triangle b=c h=r is exactly the same as the area of a circle r=r

and

2) c=2*pi*r

Neither of these assumptions are mathematically proven here, so it's a circular argument.

Best,

-Slashy

[Xannari Ferrows]

Unless I'm mistaken, this proof is circular logic. It assumes at the outset that a=pi*r^2 by setting the circumference to 26.16.

Best,

-Slashy

EDIT: You guys are ninjas or something.

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You equate 26.16 to 2*pi*r later in your proof, the radius is very much relevant. I don't see how the 26.16 is relevant to the proof, you should just leave circumference as 2*pi*r, IMO.

Ultimately you're correct, but the point of adding it was to exaggerate the fact that the two are equal, rather than just dumping the information on you.

Edited February 21, 2015 by Xannari Ferrows

[GoSlash27]

Actually, it wasn't intentionally set to 26.16. In window view, that's what the math came out like. This is also worded confusingly... Are you saying that I used the law of the circumference to help prove the law of the area? If so, then you would be right. To call that into question, however...

http://imgur.com/a/Cct4p

EDIT: You guys are ninjas or something.

Oh, I'm not calling it into question. We know that it's true. I'm just saying it's not a valid proof.

How would you mathematically prove that the area of your triangle is the same as the area of your circle? How do you know that they are exactly the same?

Best,

-Slashy

[Red Iron Crown]

I believe Slashy is referring to the "unwrapping" step of your proof; you don't demonstrate that the resultant triangle is the same area as the circle. We know it is, but you haven't proven it.

Edit: Ninja'd.

[GoSlash27]

I believe Slashy is referring to the "unwrapping" step of your proof; you don't demonstrate that the resultant triangle is the same area as the circle. We know it is, but you haven't proven it.

Edit: Ninja'd.

^ This.

Here's how I know that the area of a circle is, in fact, pi*r^2

I knocked it together as an explanation of a birthday card I gave a math nerd friend. Not actually intended as a proof, but it serves as one.

I also have a proof of pi itself floating around somewhere. Not exactly the same way it was done originally, but it does work. I'll see if I can find it.

*edit*

So here's how we skin this cat:

We start with a simple right diamond.

The vertices have a length of 1, so the hypotenuse has a length of sqrt(2). The perimeter of that shape (carried around the center) is 4*sqrt 2.

Now bisect that angle. We have now defined a new right triangle. the adjacent sides are sqrt(2)/2 and the vertex is 1.

Supposing we wanted to create 1/4 of an octagon. We would simply project our bisector to 1. The length of this new segment is 1-(sqrt(2)/2), or (2-sqrt(2)/2).

This defines a new right triangle, whose hypotenuse is sqrt((sqrt(2)/2)^2-(2-sqrt(2)2)^2).

Throwing some algebra at it, it simplifies down to sqrt(2-(sqrt(2)). Since we have doubled the number of sides of our octagon, we would multiply this by 8 (or 2^3) to find our perimeter.

We repeat the process. Bisect that angle, extend the bisector out to 1, and solve for the new right triangle.

The hypotenuse is sqrt(2-(sqrt(2+sqrt(2)) and the perimeter is this same value multiplied by 2^4)

Rinse and repeat. perimeter of a 32 sided shape: 2^5 (sqrt(2-sqrt(2+sqrt(2)))).

And again. Perimeter of a 64 sided shape:( 2^6) (sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))

This process repeats indefinitely. As we double the number of sides, the ratio of the perimeter to the radius approaches 2 pi, and we can define a calculable value for pi:

lim n->(1/0) [sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(...(2+sqrt(2n-1+sqrt(2n))))...)n-1)n)]

This is a process that can be carried put with an ink pen and parchment, or an abacus.

Simply find the square root of 2 to however many digits. Add 2. Increment a counter. Find the square root. Add 2. Increment the counter. Find the square root. Add 2. Increment the counter.

Continue this process until the last step, then *subtract* 2, increment the counter, and find the square root.

Multiply this number by 2^(counter) and *poof*: pi. And since our resultant value is, by definition, closer to pi than it is to the previous iteration, it can be proven that our result will differ from pi by less than f(n)-f(n-1). Once those 2 values differ by less than our desired margin, we can say that our result equals pi within the margin.

And since we have already proven that the square root of 2 is an irrational number, we can conclude that the sum of any irrational number and a rational number is necessarily irrational. Likewise, the square root of any irrational number is irrational.

Therefore, we have proven that pi will never repeat, no matter how many digits it's carried out to.

Posted on my FB page back in the day when I first sorted it out. I was pretty jazzed about it (I'm easily entertained)

Best,

-Slashy

Edited February 21, 2015 by GoSlash27

[Xannari Ferrows]

[GoSlash27]

https://lh5.googleusercontent.com/-hWXRpAx5vOI/VOiZBQ56NzI/AAAAAAAAA4w/VJZb3PB95qI/w1037-h360-no/Circles4.png

Aye... but you've merely restated what you've already posted.

How do you know that an unwrapped circle makes a triangle? How do you prove it?

-Slashy

[Xannari Ferrows]

Aye... but you've merely restated what you've already posted.

How do you know that an unwrapped circle makes a triangle? How do you prove it?

-Slashy

You are eager to start a debate...

I did restate what I said earlier, and it essentially proves that they are stating the same thing. So why is one accepted as proof, but not the other?

An unwrapped circle does not make a triangle, however...

The radius point does not disappear from the picture. Where would it go? It can't just vanish. In fact, imagine a second radius pointing up, and splitting the circle apart. It extends out to plant the line on the ground until it is straight. It acts as the top.

And presto! You have a triangle.

Hmm... I could let Mariyate have a turn at this if you think she can explain it in a bit more detail. I need a shower anyway.

Edited February 21, 2015 by Xannari Ferrows

[andrewas]

That follows from the fact that the circumference of a circle increases linearly with the radius. So you must get a triangle if you cut a set of concentric circles open and stack the circumferences at the same spacing as the original circles, as shown in the video linked on page one.

[GoSlash27]

You are eager to start a debate...

I did restate what I said earlier, and it essentially proves that they are stating the same thing. So why is one accepted as proof, but not the other?

An unwrapped circle does not make a triangle, however...

The radius point does not disappear from the picture. Where would it go? It can't just vanish. In fact, imagine a second radius pointing up, and splitting the circle apart. It extends out to plant the line on the ground until it is straight.

And presto! You have a triangle.

Oh, believe me I'm not. I have a congenital drama allergy

I'm just trying to clarify what is "proof".

In fact, an unwrapped circle *is* a triangle, but this assumes it to be so without proving it.

A rigorous mathematical proof will assume nothing that is not proven, and will show that for any arbitrary value the mathematical model is correct.

This demonstrates the concept (taking the posted assumptions as true), but it does not *prove* it.

Sorry if I'm ruffling your feathers; I don't mean to

-Slashy

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That follows from the fact that the circumference of a circle increases linearly with the radius. So you must get a triangle if you cut a set of concentric circles open and stack the circumferences at the same spacing as the original circles, as shown in the video linked on page one.

^ That is what I was looking for!

This must be included as part of a proof, else we cannot assume that an unwrapped circle is a triangle or the areas are the same.

Treating the "2pi" part of the circumference as a constant (K), we can show that for any radius r, the circumference is Kr. Therefore, the base of the triangle is a fixed ratio of the height for any arbitrary r. This proves that an unwrapped circle is, indeed, a triangle.

Best,

-Slashy

Edited February 21, 2015 by GoSlash27
edited for clarity

[Xannari Ferrows]

(Xany is out right now. I guess he didn't give much of a choice after all.

To tell you a secret, ;Slashy; while he was making these pictures, he was taking to me. I was really gathering information the whole time. The key component you seem to have been wanting was the fact that folds from the radius to the circumference are decreasing as you move closer to your origin. this means that an unraveled circle can be a triangle.

However, Xany told me that he expects everyone to assume the extra space is still there. What he said exactly was this:

"The conservation of energy states that matter cannot be created, nor destroyed. This principle is the same. The space between the circumference of the circle and the origin is always the same in regional area, just as a 3 dimensional object retains its' volume, even if you morph it to whatever shape you want. I don't get what this person is asking for. Proof of what? It's common sense! Even Henry demonstrates it in the video someone posted. The volume of a triangle and a circle will always be the same as long as a constant is held. In this case, the radius. The base of the triangle is the same... I don't get it. I don't need to explain anything.

Oh well. I need a shower. Mariyate, can you take over for a while?"

And then the jerk closes the page!

In short, what can be demonstrated through constants is a constant itself, therefore needs no proof.)

[Red Iron Crown]

"Common sense" has no place in a proof. If you want to call it a visualization or something, fine. But when you call something mathematical a proof, we expect every step of the way to be, well, proven.

[Xannari Ferrows]

("Common sense" Does not have place in a proof. However, what Xany was referring to was a constant, not a proof. Maybe postulate is a better word... Do I need to tell him to write some crazy 20+ picture album explaining every constant, how it functions, and what roll it plays? If everyone knows it, it doesn't need to be explained.)