Questions tagged [circles]

For questions concerning circles. A circle is the locus of points in a plane that are at a fixed distance from a fixed point.

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226
votes
24answers
48k 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 ...
186
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4answers
55k 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 $4$-...
172
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9answers
353k 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 ...
103
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8answers
59k 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 ...
93
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17answers
19k 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?
91
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4answers
9k views

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 ...
85
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4answers
24k views

A goat tied to a corner of a rectangle

A goat is tied to an external corner of a rectangular shed measuring 4 m by 6 m. If the goat’s rope is 8 m long, what is the total area, in square meters, in which the goat can graze? Well, it seems ...
78
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5answers
14k 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 ...
69
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9answers
36k 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 ...
62
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17answers
15k 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 ...
57
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2answers
3k 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.
54
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4answers
6k 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 ...
48
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4answers
9k 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 ...
46
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4answers
5k views

A conjecture involving prime numbers and circles

Given the series of prime numbers greater than $9$, we organize them in four rows, according to their last digit ($1,3,7$ or $9$). The column in which they are displayed is the ten to which they ...
43
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11answers
8k views

How was the area formula for a circle ($A = \pi r^2$) derived before the introduction of calculus?

How did mathematicians prior to the coming of calculus derive the area of the circle from scratch, without the use of calculus? The area, $A$, of a circle is $\pi r^2$. Given radius $r$, diameter $d$ ...
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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18answers
13k 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
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15answers
104k 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 ...
35
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3answers
810 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/...
35
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8answers
143k 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?
35
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5answers
3k 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 ...
31
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3answers
3k views

Why does area differentiate to perimeter for circles and not for squares?

I read this question the other day and it got me thinking: the area of a circle is $\pi r^2$, which differentiates to $2 \pi r$, which is just the perimeter of the circle. Why doesn't the same ...
29
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4answers
6k 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 ...
29
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3answers
68k views

How to determine the arc length of ellipse?

I want to determine the length of an arc from the ellipse in the picture below: How can I determine the length of $d$?
28
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10answers
30k 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.
28
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6answers
59k 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 ...
28
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3answers
4k 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 ...
27
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6answers
4k 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) ...
27
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2answers
9k 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 ...
26
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2answers
1k views

Is it possible to elementarily parametrize a circle without using trigonometric functions?

Just out of curiosity: Is it possible to parametrize a full circle or part of one with elementary functions but without using trigonometric functions? If so, what are advantages/disadvantages compared ...
25
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1answer
411 views

The generous lazy caterer

It is well-known that $n$ chords divide any convex shape into at most $\frac{n^2+n+2}2=T_n+1$ regions – the lazy caterer's sequence. For example, the pancake below is cut into seven pieces by three ...
24
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8answers
31k 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 proven ...
24
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4answers
2k 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 ...
22
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3answers
84k views

Find the coordinates of a point on a circle

I have a circle like so Given a rotation θ and a radius r, how do I find the coordinate (x,y)? Keep in mind, this rotation could be anywhere between 0 and 360 degrees. For example, I have a radius ...
21
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6answers
2k 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 $(...
21
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6answers
86k 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 ...
21
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1answer
643 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. ...
20
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10answers
30k 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 ...
19
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2answers
2k views

I found a new way of calculating pi,is that a great deal? [closed]

I am a 15 year old teen and fond of mathematics. I always try to prove mathematical theories and I tried to find how to get pi using an algorithm. Then I found out a way and made an algorithm and ...
19
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1answer
392 views

Japanese theorem, in its extended form

"Japanese theorem" is the name given to the following result (see Fig. 1 below) : if $A_1A_2A_3A_4$ is a cyclic quadrilateral, the incenters of the $4$ triangles $A_pA_qA_r$ with all triples $\{p,q,r\...
18
votes
10answers
28k 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?
18
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5answers
2k views

Determine where a point lies in relation to a circle, is my answer right?

The circle $C$ has equation $$x^2-8x+y^2+6y=24$$ I completed the square: $$(x-4)^2+(y+3)^2=49$$ Therefore the radius is 7. I want to know if the point $(9,2)$ is inside the circle. Substituting into ...
18
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6answers
817 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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5answers
3k views

Radius of a circle touching a rectangle both of which are inside a square

Given this configuration : We're given that the rectangle is of the dimensions 20 cm by 10 cm, and we have to find the radius of the circle. If we somehow know the distance between the circle and ...
17
votes
3answers
719 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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9answers
12k 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 ($\pi=3,...
16
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6answers
1k 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.
16
votes
4answers
74k 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 $...
16
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8answers
32k views

Technique for proving four given points to be concyclic?

While making my way through an exercise, I stalled on question 7: 7. Prove that the points $(9, 6)$, $(4, -4)$, $(1, -2)$, $(0, 0)$ are concyclic. The book does not provide any guidance on how to ...
16
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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.