Questions on the circle, a curve composed of points that are at a fixed distance from a fixed point.

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166
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27answers
30k views

Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers

An exam for high school students had the following problem: Let the point $E$ be the midpoint of the line segment $AD$ on the square $ABCD$. Then let a circle be determined by the points $E$, $B$ and ...
119
votes
3answers
17k views

Why can a Venn diagram for 4+ sets not be constructed using circles?

This page gives a few examples of Venn diagrams for 4 sets. Some examples: Thinking about it for a little, it is impossible to partition the plane into the $16$ segments required for a complete ...
109
votes
7answers
155k views

How many sides does a circle have?

My son is in 2nd grade. His math teacher gave the class a quiz, and one question was this: If a triangle has 3 sides, and a rectangle has 4 sides, how many sides does a circle have? My first ...
84
votes
17answers
14k views

How do you find the center of a circle with a pencil and a book?

Given a circle on a paper, and a pencil and a book. Can you find the center of the circle with the pencil and the book?
58
votes
4answers
7k views

Why is a circle in a plane surrounded by 6 other circles

When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other number? I'm ...
50
votes
2answers
2k views

Geometry problem involving infinite number of circles

What is the sum of the areas of the grey circles? I have not made any progress so far.
48
votes
17answers
10k views

Why is a circle 1-dimensional?

In the textbook I am reading, it says a dimension is the number of independent parameters needed to specify a point. In order to make a circle, you need two points to specify the $x$ and $y$ position ...
41
votes
4answers
3k views

Do circles divide the plane into more regions than lines?

In this post it is mentioned that $n$ straight lines can divide the plane into a maximum number of $(n^{2}+n+2)/2$ different regions. What happens if we use circles instead of lines? That is, what ...
40
votes
17answers
3k views

How to create circles and or sections of a circle when the centre is inaccessible

I am doing landscaping and some times I need to create circles or parts of circles that would put the centre of the circle in the neighbours' garden, or there are other obstructions that stop me from ...
39
votes
7answers
17k views

Why is the derivative of a circle's area its perimeter (and similarly for spheres)?

When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle. Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is ...
39
votes
3answers
2k views

Cutting up a circle to make a square

We know that there is no paper-and-scissors solution to Tarski's circle-squaring problem (my six-year-old daughter told me this while eating lunch one day) but what are the closest approximations, if ...
27
votes
3answers
2k views

Have I made a straight line, or a circle?

(Disclaimer: I'm an engineer) Hi everybody, I found this “riddle” posted on the internet: It's meant as a joke, but I do think it deserves an answer :) A bit of background: the orange and blue ...
26
votes
8answers
4k views

How to find center of a circle from only an arbitary arc of that circle

How to find the center of a circle with given an arbitrary arc. we only have the arc nothing else. Is there any known equation or way to complete the circle.
26
votes
13answers
29k views

Calculus proof for the area of a circle

I was looking for proofs using Calculus for the area of a circle and come across this one $$\int 2 \pi r \, dr = 2\pi \frac {r^2}{2} = \pi r^2$$ and it struck me as being particularly easy. The only ...
25
votes
4answers
1k views

Two circles inside a right angled triangle!

The other day I was playing with Ms Paint drawing circles here and there - I coincidentally drew a circle inside a right angled triangle which I already drew. Strangely A problem struck to my mind ...
24
votes
4answers
1k views

Finding an invisible circle by drawing another line

A friend of mine taught me the following question. He said he found it in a book a few years ago. Though I've tried to solve it, I'm facing difficulty. Question: You know on a plane there is an ...
23
votes
5answers
1k views

Trying to understand why circle area is not $2 \pi r^2$

I understand the reasoning behind $\pi r^2$ for a circle area however I'd like to know what is wrong with the reasoning below: The area of a square is like a line, the height (one dimension, length) ...
23
votes
5answers
1k views

Did Euclid prove that $\pi$ is constant?

Pi is defined the ratio of the circumference of a circle to its diameter, but of course different circles have different circumferences and diameters, so in order for it to be well-defined we need to ...
22
votes
2answers
2k views

Divide circle into 9 pieces of equal area

I'd like to divide a unit circle disk into nine parts of equal area, using circle arcs as delimiting lines. The whole setup should be symmetric under the symmetry group of the square, i.e. 4 mirror ...
20
votes
1answer
430 views

Infinite staircase to a circle

Suppose you start at $(0,0)$ on the unit disc and repeat the following procedure again and again: Face east and walk half-way to the circumference. Face north and walk half-way to the circumference. ...
18
votes
6answers
734 views

Is this 3D curve a circle?

The following is a curve in $3$ dimensions: $$\begin{eqnarray} x & = & \cos(\theta) \\ y & = & \cos(\theta - \pi/3) \\ z & = & \cos(\theta - 2\pi/3) \end{eqnarray}$$ Is the ...
17
votes
6answers
429 views

Why is the area of the circle $πr^2$? [duplicate]

I searched many times about the cause of the circle area formula but I did not know anything so ... Why is the area of the circle $\pi r^2$? Thanks for all here.
17
votes
3answers
343 views

What is the largest circle that fits in $\sin(x)?$

Imagine dropping a circle into the trough of $\sin(x)$. Would it reach the bottom or get wedged between two points on the curve? Depends on the size of the circle. So, what is the radius of the ...
15
votes
6answers
13k views

A circle with infinite radius is a line

I am curious about the following diagram: The image implies a circle of infinite radius is a line. Intuitively, I understand this, but I was wondering whether this problem could be stated and ...
14
votes
6answers
747 views

Why do we use the Euclidean metric on $\mathbb{R}^2$?

On the train home, I thought I would try to prove $\pi$ is irrational. I needed a definition, so I used: $\pi$ is the area of the unit circle. But what is a circle? A circle is the set of tuples ...
14
votes
1answer
2k views

How many circles to cover 2 times bigger circle?

How many circles (radius – r) are needed to cover circle which radius is 2 times bigger (radius – 2r). I think we need to use area which is $S=\pi R^2$ but I don't really know what to do
13
votes
10answers
3k views

Finding circumference without using $\pi$

If the area of a circle is $254.34\ldots\text{ cm}^2$ it has a diameter of $18\text{ cm}$, is it possible to find the circumference without using or making the irrational constant Pi ...
13
votes
3answers
656 views

Covering the plane with disks

How to prove that it is impossible to cover the plane with disks? /The disks are closed disks and two disks can meet (at most) at only one point (obviously on the border)./ Thank you very much in ...
13
votes
1answer
228 views

A question on circles

Given that a lamp-post can light a surrounding circle of radius 100 m, what is the minimum number of such lamp-posts required to light a circular ground of radius 1000 m.
12
votes
10answers
718 views

How is the value of $\pi$ ( Pi ) actually calculated?

When I was a child I was taught $\pi$ (Circumference/Diameter) is an irrational number and can be approximated to $22/7$ but $= 3.(142857)(\ldots)$. But where does this value comes from? In ...
12
votes
4answers
2k views

Is it possible to divide a circle into $7$ equal “pizza slices” (using geometrical methods)?

Or is it possible to divide a circle into n equal "pizza slices" (I don't know how to call these parts, but I think you'll know what I mean), where n hasn't got a common divider with $360$? Or are the ...
12
votes
5answers
18k views

Parametric Equation of a Circle in 3D Space?

So, my dilemma here is... I have an axis. This axis is given to me in the format of the slope of the axis in the x,y and z axes. I need to come up with a parametric equation of a circle. This circle ...
12
votes
1answer
2k views

A hard geometry problem on circles

I found this problem on a website and I couldn't do anything. Do you have any ideas, hints? Edit: If I say $$\frac { { \partial }^{ 2 }f }{ \partial { a }^{ 2 } } +\frac { { \partial }^{ 2 }f }{ ...
12
votes
4answers
2k views

How to equally divide a circle with parallel lines?

How can I "draw" $n$ parallel lines in such a way that they will divide a circle (disc) in $n+1$ equal areas ?
12
votes
2answers
2k views

Proving collinear points

This problem is so hard that I cannot figure it out. I hope you guys can give me a small push on how to tackle this problem, as I have been thinking about this for, like a week. Here's the problem: ...
12
votes
1answer
560 views

How does one calculate the product of $\tan 1^{\circ} … \tan 45^{\circ}?$

I have seen a question asked on yahoo asking to find the value of $\tan 1^{\circ} \cdot \tan 2^{\circ} \cdot \dots \cdot \tan 45^{\circ}$ (in degrees) I have seen various results concerning ...
12
votes
1answer
854 views

A geometry problem seeking for proof

Circle $\odot O_1$ is tangent with circle $\odot O_2$ at $P$. Two tangent lines $AE$ and $AF$ of circle $\odot O_2$ meets circle $O_1$ at $B$, $G$ and $C$, $H$, respectively. $D$ is the in-center of ...
12
votes
2answers
403 views

6 point lying on a common circle

$Z$ is an interior point of segment $XY$. Three semicircles are drawn over segments $XY$, $XZ$ and $ZY$ on the same side. The midpoints of the arcs are $M1$, $M2$ and $M3$ respectively. A circle ...
12
votes
2answers
375 views

Smallest inradius in a triangle

Inside triangle ABC there are three circles with radius $r_1$, $r_2$, and $r_3$ each of which is tangent to two sides of the triangle and to its incircle with radius r. All of $r$, $r_1$, $r_2$, and ...
11
votes
7answers
13k views

a circle graph is not a function?

I'm a little confused by the rule: If you draw a vertical line that intersects the graph at more than 1 point then it is not a function. Because then a circle like $y^2 + x^2 = 1$ is not a function? ...
11
votes
1answer
1k views

Diffeomorphism group of the unit circle

I am given to understand that the group of diffeomorphisms of the unit circle, $\operatorname{Diff}(\mathbb{S}^1)$, has two connected components, $\operatorname{Diff}^+(\mathbb{S}^1)$ and ...
11
votes
5answers
28k views

How can I find the points at which two circles intersect?

Given the radius and $x,y$ coordinates of the center point of two circles how can I calculate their points of intersection if they have any?
11
votes
6answers
18k views

Area of intersection between two circles

Suppose you have 2 circles that intersect each other in such a way that each circle passes through the other's center. What is the area between the circle(or common area) i.e. area between the centres ...
11
votes
1answer
153 views

Is a line just an infinitely large circle?

A line is infinite, right? Well, if $-\infty = \infty$, then a line is an infinitely large circle. (Does this have something to do with $1/0$?) It seems wrong, is it? Could I disprove it? How ...
11
votes
2answers
233 views

Task “Inversion” (geometry with many circles)

Incircle $\omega$ of triangle $ABC$ with center in point $I$ touches $AB, BC, CA$ in points $C_{1}, A_{1}, B_{1}$. Сircumcircle of triangle $AB_{1}C_{1}$ intersects second time circumcircle of $ABC$ ...
11
votes
1answer
334 views

Is there a way to represent the interior of a circle with a curve?

As you already know, the interior of a circle is represented by an inequality. For example, $$x^2+y^2\leq1$$ for the unit circle. Today I was thinking by myself and I wondered if there is a curve ...
11
votes
2answers
454 views

Is the figure the circumference of a unit circle?

A friend of mine taught me the following question. I've never heard such a strange and interesting question! Qustion: Supposing that a figure $S$, which is constituted by points, satisfies the ...
11
votes
3answers
118 views

How can the centers of these 5 related circles be specified as a formula?

This is my first time posting in this forum, so please forgive me if my question is too involved or if I've posted it in the wrong area. I hope someone finds it interesting enough to try their hand at ...
10
votes
3answers
4k views

What is the probability that the center of the circle is contained within the triangle?

Consider the triangle formed by randomly distributing three points on a circle. What is the probability of the center of the circle be contained within the triangle?
10
votes
5answers
7k views

Calculate the area of the crescent

I found this problem on a thread on Stack overflow where it was posted as "job interview question". Unfortunately I cannot find the question. But I saved the picture and just cannot figure it out. ...