For questions conserning circles. A circle is a curve composed of points in a plane that are at a fixed distance from a fixed point.

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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 ...
140
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3answers
25k 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 $4$-...
129
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9answers
210k 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 ...
86
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17answers
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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?
80
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4answers
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The “pepperoni pizza problem”

This problem arose in a different context at work, but I have translated it to pizza. Suppose you have a circular pizza of radius $R$. Upon this disc, $n$ pepperoni will be distributed completely ...
66
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5answers
9k 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 numbers? I'm ...
59
votes
9answers
32k views

Why is $\pi $ equal to $3.14159…$?

Wait before you dismiss this as a crank question :) A friend of mine teaches school kids, and the book she uses states something to the following effect: If you divide the circumference of any ...
58
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17answers
11k 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 ...
53
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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.
53
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8answers
29k 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 ...
46
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4answers
4k 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 ...
43
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2answers
3k views

Are similar circles really a thing?

I'm a fifteen year old who is currently studying circle geometry (if that is the appropriate term) and our teacher stated that concentric circles are similar. I thought about this, and it doesn't make ...
41
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3answers
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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 ...
40
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17answers
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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 ...
31
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3answers
533 views

Can all circles of radius $1/n$ be packed in a unit disk, excluding the circle of radius $1/1$?

This problem occurred to me when I came across a similar problem where the radii were taken over only the primes. That question was unanswered, but it seems to me infinitely many circles of radius $1/...
29
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3answers
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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 ...
28
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13answers
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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 ...
27
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4answers
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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 ...
27
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5answers
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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 ...
26
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9answers
10k 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
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5answers
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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) ...
25
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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 ...
24
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2answers
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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
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1answer
493 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. ...
19
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6answers
17k 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 ...
18
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6answers
763 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
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9answers
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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 ($\pi=3,...
17
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6answers
835 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
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3answers
404 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 ...
16
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5answers
29k 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 ...
15
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6answers
34k views

Area of intersection between two circles [duplicate]

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 ...
15
votes
7answers
365 views

Locating three sets of collinear points

Given any three distinct points $A,B,C$ and a circle $C(O)$, construct points $D,E,F$ on the circle such that $A,D,E$ are collinear, $B,E,F$ are collinear and, $C,F,D$ are collinear. One such ...
14
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6answers
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How is the area of a circle calculated using basic mathematics?

Area of a circle is addition of circumference of layers of a onion. If n is radius of a onion then area is $$ A = 2 \pi \cdot 1 + 2 \pi \cdot 2 + 2\pi \cdot 3 + \ldots + 2 \pi \cdot n $$ which $$ =...
14
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6answers
937 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
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4answers
1k views

What is the area of the circle?

In the following diagram, $AB = 4$ and $AC = 3$. What is the area of the circle? I can't find any way to solve this.
14
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7answers
53k 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?
14
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1answer
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How many circles to cover 2 times bigger circle?

How many circles (radius $r$) are needed to cover circle whose 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
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7answers
1k views

Length of a Chord of a circle

I was wondering about the possible values that the length of a chord of a circle can take. The Length of a chord is always greater than or equal to 0 and smaller ...
13
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10answers
3k 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 ...
13
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4answers
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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 ...
13
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3answers
773 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
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1answer
232 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.
13
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1answer
3k 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 }{ \...
13
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5answers
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 ?
13
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1answer
352 views

How big is my pizza, if I know its slices' sizes?

I bought a box of frozen pizza: eight slices, baked and then frozen, stacked in a box. The packaging assured me that I was holding an 18-inch[-diameter] pizza. That got me thinking: how do I know they'...
13
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2answers
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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
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7answers
23k 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? ...
12
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4answers
6k 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?
12
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3answers
39k views

How do I calculate the intersection(s) of a straight line and a circle?

The basic equation for a straight line is $y = mx + b$, where $b$ is the height of the line at $x = 0$ and $m$ is the gradient. The basic equation for a circle is $(x - c)^2 + (y - d)^2 = r^2$, where $...
12
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3answers
26k views

How to determine the arc length of ellipse?

I want to determine the arc length of a ellipse. So what data should I know ? And what law should I use ? For example I have this ellipse on picture below: How can I determine the $d$ length of ...