Is a Circle Bigger than a Square?

[Splice]

If you get a square, and put a right angle triangle in it so that each point of the triangle touches the outline of the square, that right angle triangle is half the size of the square, right?

Well if you put a right angle in a circle, and make it so the points of the triangle touch the outline of the circle, the left over space in the circle is BIGGER than the triangle.

So the same size right angle triangle would take up a bigger proportion of it's relative square than it would of it's relative circle.

Seeing as a circle or square can be any size, we can use this Triangle as an object which we can relate and measure by, thus answering the age-old question

Circles are bigger than Squares.


RATINGS AND REVIEWS:

"10/10 splice" - EA
"8/10 troll thread" - Summoner
"Very good topic [for] trolling" - Nixernator
"It's very good" - Grim Tuesday
"That's stupid" - My Mum
"Ah..." - Corpsecreate

thankyou everyone!

 

[Pantsmann]

Seeing as a circle or square can be any size, we can use this Triangle as an object which we can relate and measure by, thus answering the age-old question

what? that's a purely arbitrary unit of measurement. how about we use the SI unit of m^2 as what we use to relate and measure by. lo and behold - squares and circles are exactly the same size

 

[Splice]

How is it arbitrary? If you used the same size triangle, and use this method, the circle will turn out to be bigger.

You can make a square and a circle the same size, you can make them any size all the way to infinity. That is obviously not going to help us find an answer.

Using the relative unit of the Right Angle Triangle, there is a way.

Circles are bigger than Squares.

 

[xXArrowXx]

wtf r u guys talking about..

define this size u speak of... u mean same area? or diameter = side of square :S

really depends where u put the triangle.... for one thing.

if the circle and the square have the same area. TTr^2 = s^2
and area of triangle is 0.5*base*height.
then the circle occupies less area than the square.. so yeah the circle has more left over room..and the triangle in the square takes up more space than the circles.. but they both have the same amount of area in their respective shape (circle and square)
so i dont know what u trying to prove.
its just u cant fit a big enough right angle triangle in a circle.
the maximum u can get out of a circle is an equilateral triangle.

 

[asianaussie]

right angle triangle is admittedly a less arbitrary measure but also a far less logical and appliacble one than standard units of measure

also you haven't shown exactly what sort of 'size' you're referring to - seems to be a convoluted mess of how a triangle interacts with the nature of a shape's outline

yawn

 

[Splice]

That gets you nowhere arrow

because of the different... shape ...of a circle and a square, if you make them the same size, the triangle obviously won't fit in both if it's the same size!

You are TRYING to make them the same size, which is why you are coming to your conclusion.

Realise it's about touching the perimeter, not about keeping the triangle the same size, because that's useless, it won't fit in both, that'd get us nowhere due to aforementioned reasons concerning the difference in shape of square and circle.

@aa: it seems pretty easily applicable to me lol. size = area, and yes it is dictated by triangular interaction, because all the other methods are obviously created with the thought in mind that circles and squares CAN be the same size, but not thinking about proportion, which can only be decided when another shape is involved in a manner such as this.

Circles are bigger than Squares.

 

[Ghostbone]

All this shows is that if you put a square inside a circle so that the square completely fits inside the circle, the circle with have a larger area.

No duh

If you did the same thing but with a different shape you'd get a different result. (e.g., more space is left over in the square using a semi-circle)
All that would show is that if you put a circle inside a square, the square will have a larger area.

Circles are not bigger than squares.

 

[SummonerAU]

8/10 troll thread

 

[Splice]

ghost, what shape with less sides than a square and more than a circle (so that it is reasonable to measure by) gives you the result a square is bigger?

Circle Triangle Square

They are the three basic shapes and they can be measured against one another to see

If use Hexagons etc. it doesn't work because they have more sides than a square and a circle
If we could do that we'd never get anywhere

Support progress instead of trying to save what you think you know
if everyone did that you'd still think the world was flat.

 

[xXArrowXx]

big circle is big.

 

[...Ellipsis...]

What he is saying is true, if you take a circle with a larger surface area than a square, it will be bigger than that square. Simple maths guys.

 

[Splice]

and if you get a square bigger than a circle, the square will be bigger.

Do you see why thats not an elaborate method?
That's not what I am using.

The relative Right-Angle-Triangle is the only way to get an answer that doesn't go around in these stupid circles.

Move forwards guys.

Even if you get a hexagon or something that's not a triangle, it's worth noting that the triangle still takes up more room in these shapes than it does in the circle.

The circle is the largest shape.

 

[...Ellipsis...]

Yeah but consider if you had a sphere. When you put a triangle in the sphere it only takes up a tiny proportion as the sphere has a whole other dimension. Therefore the sphere is the largest shape.

 

[xXArrowXx]

wtf r u guys on about...
u dont need triangles at all.. u can work out which is bigger with a dimension. >.> as in the radius of the circle and a side of the square... there is no need for triangles. all u do by putting a triangle in another shape is seeing the proportion that it occupies :/

also btw the biggest triangle (the one that takes up the most area) u can fit in a circle is a (1,0),(0,1),(-1,0) triangle. not a right angled triangle...

 

[Splice]

yeh but the triangle we put in the square is a Right Angle.

That's a different shape you got there.

@... : Yes, I guess a sphere would be the biggest shape. Let me rephrase
The circle is the largest 2-D shape

 

[Corpsecreate]

Circle has infinite sides/corners. A circle by definition is an equilateral polygon with n sides as n -> infinity. Theoretically speaking, a circle doesn't actually exist just as 1/0 doesn't exist. A circle is the shape that is formed by evaluating a limit.

I also don't understand what Splice is trying to prove.

Circle area = pi*R^2
Square area = L^2

Saying a circle is bigger than a square would depend on the dimensions no? I think what you mean to say is that a circle is the "most optimal" shape. A circle will result in the same area as another shape whilst having a smaller perimeter.

For instance, a Circle with Area pi would have radius 1 and perimeter 2pi = 6.283
A square with Area pi would have length sqrt(pi) and has perimeter 4*sqrt(pi) = 7.09

This is easily proven for any area by getting L in terms of R and then comparing derivatives:

pi*R^2 = L^2 ----> L = R*sqrt(pi)

Perimeter of Circle = 2*pi*R
Perimeter of Square = 4L = 4R*sqrt(pi)

dP/dR (circle) = 2pi for all R
dP/dr (square) = 4*sqrt(pi) for all R

2pi < 4*sqrt(pi) for all R.

Circle is more optimal than square.

 

[ciaza]

What if I place an oval inside the circle? And then I placed the same oval inside the square? The oval takes up more area of the circle than it does the square! Ergo, squares are bigger than circles.

The main point is that the right-angled triangle is biased to the circle just as an oval is a biased towards a square.

 

[xXArrowXx]

shapes dont have set dimensions assigned to them.
still dont see any proof to back up your outlandish logic.

putting shapes in shapes isn't proving anything >.>

 

[ciaza]

I knew I should have used my sarcasm font.

 

[SummonerAU]

Is a nudge as good as a wink to a blind bat?

 

[Ghostbone]

What if I place a circle inside a circle and the first circle takes up 100% of the area of the second (i.e: the 2 circles are the same size)? And then I placed the same circle inside the square? Oh ****, the circle doesn't take up all of the square! Ergo, squares are bigger than circles.

The main point is that the right-angled triangle is biased to the circle just as a circle is a biased towards a square.

This, and Splice, why use an irregular shape, a right-angled triangle, why not an equilateral triangle? You get the same result, but still, your method is biased.

So basically every method is biased, and what you're trying to prove makes no sense.

 

[Splice]

Ghost, you use right angle triangles in to find the cosine, sine and tangent of 45, 30 and 60 degrees. These also correspond with the unit circle.

Summoner, wouldnt a nudge obviously be better than a wink for a blind bat?

 

[EverAlert]

10/10 splice

 

[xXArrowXx]

Ghost, you use right angle triangles in to find the cosine, sine and tangent of 45, 30 and 60 degrees.

i though the inner angles of a triangle suppose to add up to 180 degrees :S
u can use cosine and sine and tan on any triangle not just right angled ones...

 

[Splice]

Arrow, I realise you may not have seen this.

Please read:
http://www.themathpage.com/atrig/30-60-90-triangle.htm

And more simple:
http://en.wikipedia.org/wiki/Special_right_triangles

 

[Corpsecreate]

I pose a question, what is the area of an equilateral n-th sided polygon with sides of length s?

 

[...Ellipsis...]

I pose a question, what is the area of an equilateral n-th sided polygon with sides of length s?

http://lmgtfy.com/?q=area+of+an+equilateral+polygon

 

[Ghostbone]

Ghost, you use right angle triangles in to find the cosine, sine and tangent of 45, 30 and 60 degrees. These also correspond with the unit circle.

Summoner, wouldnt a nudge obviously be better than a wink for a blind bat?

So? Your method is still biased. Right angled triangles are just used to make things simpler lol, they're not meant to be a unit of measurement like you're using them for.

 

[xXArrowXx]

i see how u got 45,45, 90 and 30,60,90
and i see how u use them because u get easy numbers to work with.
i dont see how u choose which one and how that relates to anything
the triangle takes up a proportion of the shapes area. slightly less in circles.
doesn't make the circle bigger. the circle is the same size as a square. however u can fit a bigger right angle triangle in a square.

 

[Splice]

. the circle is the same size as a square. however u can fit a bigger right angle triangle in a square.

The circle and the square have to be the same size for you to fit a bigger triangle in a square. But if the Circle was bigger, the same size right angle triangle would fit into both.

I can't believe you guys aren't getting this...

 

[Ghostbone]

The circle and the square have to be the same size for you to fit a bigger triangle in a square. But if the Circle was bigger, the same size right angle triangle would fit into both.

I can't believe you guys aren't getting this...

Well, all that shows is that a circle needs to be bigger than a square to fit the same size right angled triangle in it.

Not that circles are bigger than squares.

It doesn't make sense to make the connection.

Edit: Also missed something you said earlier, saying circles are the larges 2-D shape.
Anything irregular is larger by your method.

Also, why do we have to use triangles? why not pentagons, hexagons, etc.?

 

[xXArrowXx]

u r just scaling the size of the circle up
so of course it will be bigger. (more area)

so looking it another way. u put a hexagon in a circle and a square.
the square cant fit the hexagon in as big as the circle. so u scale up the squares size so that the hexagons are the same size. and the square is now bigger than the circle.

 

[asianaussie]

I can't believe you guys aren't getting this...

 

[MTGod]

Circles are the biggest 2D shape
v
Spheres are the largest 3D shape
v
Spheres are larger than cylinders
v
Our brain receives information via cylindrical tubes sending electrical pulses
v
Our brain can't comprehend spheres.

 

[xXArrowXx]

I saw that movie.
what was in that sphere i couldn't comprehend