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wHEN YOU SHRINK THEREST OF THE CIRCLE, THE REMAINDER SHOULD FIT EXACTLY ONTO THE CIRCLE.

duh? shouldn"t it be same then?

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I wonder if pi can be a recurring number (each "line" is maybe for around 100000000000000000000000000000 digits.)

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Pi might be rational at MANYMANYMANYMANY digits

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Who discovered π and how did he/she found it out?

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How is pi discovered/created and who discovered/created pi? Was the discovery/creation an accident or was it intentional?

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Maybe a perfect circle requires a line thinner than the yoctometre.

Thinnest measurement on earth. Now even thinner

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Yup. It works for infinite number of circles.

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I think my "Think" part is wrong. I want to know the REAL reason

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I wonder who discovered  in the first place...

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Circles are very special

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I wonder if there are other

circle tricks.

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@HM

It is actually 1.24 trillion. Discovered by a supercomputer

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How do they find out so many digits?

(the computers cannot calculate so many digits)

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Pi was first rigorously calculated by one of the greatest mathematicians of the ancient world,

Archimedes of Syracuse (287-212 B.C.).

Archimedes was so engrossed in his work that he did not notice that Roman soldiers had taken the

Greek city of Syracuse. When a Roman soldier approached him, he yelled in Greek,

“Do not touch my circles!” The Roman soldier simply cut off his head and went on his business.

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If the three circles (or any other circle) is out, the circumference will not be the same anymore

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I wonder if how people discovered π?

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This happens basically because just imagine that there is one big circle in the big circle. They have the same size (area, diameter, radius, ...)So obviously it would have the same circumference

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I think that no matter how many digits of pi we find, there is always still some more. Thus, I feel that people should just stop looking for digits of pi.

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Maybe because the diameter of the smaller circles share the one of the bigger circle

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I think this should be considered i think

There is a theory call the Intersecting chord theorum

NOTE: nothing to do with Pi

I think.

There is also a thery call Tangent theory,

Whereby a line that touches a point of the circle once and only once.

Though I think it is impossible, because...

1) 3D would come back to the point especially when in a concaved 3D shape.

Though i think there is a name to it

I think.

2) In a super pixelated figure, it is either it cannot be hand-drawn, or that the line would touch to points of the (pixelated) circle.

I think

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How many digits of pi are discovered by now?

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I wonder why that happens?

Is it something about pi?

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I wonder if there is any relationship of the area of circles too.

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I think that this is true because circles are magical and have certain special properties!(like unicorns!)

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I wonder why circles are so interesting. Basically, everything about circles are relating to the most basic parts of a circle.

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As the the sum of the diameter of the small circles is the same as the big circle, the circumference will also be the same.

(The multiplication rule: if a+b+c = x

then (a+b+c)*y = x*y

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4)The sum of the circumference of the circles along the

diameter of the smaller circles along the diameter is twice the diameter,

as their diameters add up to the big circle's diameter. The

diameter is half of the circumference, so we would get the same

result.

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I wonder whether the diameter has something to do with the sum and circumference of the big circle being the same.

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I wish there was a simpler proof.

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I wonder if circles are so "awesome", could there be anymore stuff that we could do with circles?

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Why must circles be so COMPLICATED?

Didn't anybody think of something simpler?

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sum of the circumference of all the smaller 3 circles is the circumference of the bigger circle.(?)

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The sum of the circumference of the 3 circles add up to the circumference of the big circle.

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I think the smaller circles can

actually be expanded to

become the circumference of

the biggest circle.

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Why would pi be rational?

It's decimal places are not rational at all!

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What is circumference based on

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The circumference of the big circle being the same as the sum of the circumference of the small circles has something to do with them sharing the diameter of the big circle.

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For Darren: Ya...

Because of the diameter. The diameter sum of the small circles = the diameter of the large circle.

pi times diameter is circumference.

Small circles have same diameter sum as diameter of large circle, so diameter is the same.

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i think I know why the sum of all the small cicle's circumference=

the circumfernece of the big cicle.

The formula for finding the circuference is: pi times diameter

since the sum of the all the small circle's diameter is equals to the

diameter of the big cicle's diameter, and the pi is always the same,

this prove that the sum of all the small cicle's circumference=

the circumfernece of the big cicle. PROVE IT! Yeah.

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Well the small circle's diameter add up to the diameter of the

the big circle...

( diameter of small circle 1+diameter of small circle 2)x π= Diameter of big circle x π

Yes, my post is lame. I don't have any other thing to say

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Since the total diameters of

the small circles are equivalent to the

diameter of the big circle, the two values will be

equal, since the formula for the

circumference of a circle is 2 x pi x radius

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Does it have something to do with the diameter of the circles added together?

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The circumference of the small circles is equal to the circumference of the big circle

But if the small circles overlap, this doesn't happen.

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The sum of circumference

of the small circles is the

same as the circumference

of the big circle

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The smaller circles' sum add up to

the circumference of the biggest circle.

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I think the circles are actually half of

the big circle.

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The sum of the three circles is the equivalent to the BIG circle.

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The sum also= big circle in two

triangle, except if you really mess up with the triangle

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circles are weird

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Radius of one small circle = r
Circumference of one small cirle = 2πr
Sum of circumferences of 4 small circles = 4*2πr = 8πr
Radius of the big circle = R
Circumference of the big circle = 2πR
As per the question : 2πR = 8πr , => R/r = 4/1
Area of the Big circle = A = πR²
Area of the small cirle = a = πr²
Hence , A/a = R² / r² = (R/r)² = (4/1)² = 16 / 1

Okay i searched online

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This circle stuff is pretty complicated. Does circumference have a part in determining area

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Maybe pi is rational?

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I wonder

Would this pattern continue to an infinite number?

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The circrumference of the big circle is the same as the circumference of the smaller circles added together.

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The circumference of the small circles is the same as the big circle.

That's so cool!

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No matter how you change the size of any circle, as long as

the circumference of the circles are touching each other, the

sum of the circumferences of the small circles and the

circumference of the big circles will be equal.

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I wonder why each part of the circle like the diameter and circumference has special properties. Miracle of mathematics

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I think Darren should simplify his proof

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I see that the sum of the circumferences of the smallcircles is equal to that of the big circle!

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It works for more than 4 circles too... Right?

Edit: Yes, thanks to the formula

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the sum of the 3 circles' circumfrance is the same as the circumfrance of the big circle.

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The sum=big circle, cool!

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I think

I think that the sides of the smaller circle can be moved such that they can fit the circumference of the larger circle

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The circumfrence of small adds up to the circumfrence of the big circle

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I see the sum of the circles=the circumference of the circle.

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I see...

The circumference of the small circles add up to the circumference of the big circle.

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Length of circumference depends on pi

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I wonder why circles have such interesting properties

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when the sides are 'bent' to fit the half circumference,they fit.

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Proof:

Let diameter of Circle 1 be A

Then diameter of Circle 2 is B

Diameter of circle 3: C

Big circle is A+B+C

Circumference of Circle 1 be π(A)

Circumference of Circle 2 is π(B)

Circumference of circle 3: π(C)

Big circle circumference is π( A+B+C)

= π(A) + π(B)+π(C) [simple algebric ]

Note that the sum of circumference of Circle 1 , Circle 2, Circle 3

= π(A) + π(B)+π(C)

(Obviously,) sum of circumference of Circle 1 , Circle 2, Circle 3

= Circumference of BIG CIRCLE

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Circumference of big circle = Sum of circumferences of small circles

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The circumference of the tiny circles add up to the big circle's circumference.

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Well, the circumference is round and surrounds the entire circle

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The sum of the smaller circles in the bigger circle is

actually the circumference of the big circle.

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Based on what you have done on Geometric Sketchpad so far on the investigation of circumference, post 3 separate wall-posts.

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Based on what you have done on Geometric Sketchpad so far on the investigation of circumference, post 3 separate wall-posts.

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Remember to indicate your name and title. Put your post under the correct category!

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What does it make you wonder?

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What do you think is going on?

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What do you see?