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

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56
votes
4answers
6k 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 ...
93
votes
7answers
131k 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 ...
9
votes
5answers
12k 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 ...
9
votes
5answers
20k 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?
10
votes
3answers
3k 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?
11
votes
5answers
16k 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
4answers
652 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 ...
7
votes
1answer
617 views

How to draw ellipse and circle tangent to each other?

The circle $c$ is given as are the points $A$ and $B$, which are ellipse's foci. Now I need to construct the ellipse that is tangent to the circle $c$ such that the points $A$ and $B$ are its foci. ...
14
votes
6answers
11k 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 ...
12
votes
10answers
566 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 ...
3
votes
4answers
3k views

Is it possible to build a circle with quadratic Bézier curves?

i'm searching for a curve type with a minimum of functionality and maximum of usability. I run into quadratic Bézier curves and i wonder, if its possible to draw a circle with it.
2
votes
2answers
5k views

Find the differential equation of all circles of radius a

Can someone please post a detailed step-by-step procedure. Given the circle with a radius a, what is the differential equation of the circle.
7
votes
2answers
2k views

Numbers of circles around a circle

"When you draw a circle in a plane of radius 1 you can perfectly surround it with 6 other circles of the same radius." BUT when you draw a circle in a plane of radius 1 and try to perfectly surround ...
3
votes
0answers
554 views

Circle Packing Algorithm

I have question related to circle-packing. The problem is to find the circle of minimum radius enclosing four non-overlapping circles of arbitrary radius. I have to write a program in C for this ...
7
votes
4answers
384 views

Area of intersection between 4 circles centered at the vertices of a square

The centers of four circles are at the vertices of a square of sidelength 100m. Each circle has the radius of 100m. Which is the area of their intersection?
6
votes
2answers
9k 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 ...
3
votes
8answers
505 views

Find the approximate center of a circle passing through more than three points

Consider n point $(x_1,y_1), (x_2,y_2),\ldots, (x_n,y_n)$. For $n = 3$ it is easy to find the center of the circle passing through the three points. I wanted find the approximate center of the ...
2
votes
2answers
353 views

Proof: Invariant angle measure - same result for any circle drawn.

Below I have quoted Wikipedia. I am particular interested in the statement: The value of $\theta$ thus defined is independent of the size of the circle: if the length of the radius is changed ...
8
votes
2answers
8k views

Finding the intersecting points on two circles

Given 2 circles on a plane, how do you calculate the intersecting points? In this example I can do the calculation using the equilateral triangles that are described by the intersection and centres ...
2
votes
1answer
3k views

Calculating circle radius from two points on circumference (for game movement)

I'm designing a game where objects have to move along a series of waypoints. The object has a speed and a maximum turn rate. When moving between points p1 and p2 it will move in a circular curve ...
2
votes
2answers
2k views

Counting number of distinct regions with intersecting circles

Given $n$ circles of possibly different radii, how many distinct regions can there be? For small $n$, I can work it out with pictures. (I'm pretty sure $n=4$ can yield 13 distinct regions, but not ...
1
vote
1answer
254 views

Euclidean Circle Geometry Problem

Let $\Gamma_1$ and $\Gamma_2$ be two non overlapping circles with centers $O_1$ and $O_2$ respectively. From $O_1$, draw the two tangents to $\Gamma_2$ and let them intersect $\Gamma_1$ at points $A$ ...
0
votes
3answers
4k views

Circle and Line segment intersection

I have a line segment (begin $(x_1,y_1)$, end $(x_2,y_2)$, with $D=5$, let’s say) and a circle (radius $R$, center $(x_3,y_3)$) How can I check that if my line segment intersects my circle? picture ...
2
votes
1answer
78 views

Pdf for distance between two uniform random points in a circle

This is my first post in the group and I would be very thankful for any help. I am trying to develop a probability distribution for a performance analysis in my thesis. I am trying to look in to ...
2
votes
1answer
136 views

What are the subsets of the unit circle that can be the points in which a power series is convergent?

Let $A\subset\Bbb C$ be a subset of the unit circle. Consider the following condition on $A$. Cond. There exists a sequence $\{a_i\}_{i=1}^\infty$ of complex numbers such that $$\sum_{n=1}^\infty ...
2
votes
1answer
214 views

Area of a portion of an arbitrarily-placed circle?

I have a circle that's off-center, but I want to find out the area of the part of the circle in the positive x and y region. Not sure how to do this because of the multiple variables involved.
0
votes
1answer
72 views

what are the various fields in which circle is treated as infinite sided regular polygon?

What are the various fields in which circle is treated as infinite sided regular polygon? What I actually mean is , "can u suggest me some applications where circle is treated as infinite sided ...
0
votes
1answer
254 views

How to calculate the coordinates of the middle point of a given arc?

Does anybody know how to solve this problem? I am trying to calculate the green sides of this triangle: I know / have : the arc length, the arch base, the radius, and the h (distance from the red ...
0
votes
2answers
1k views

Angle of reflection off of a circle?

I've made simple 2D games in the past using mostly just squares. If an object collided with another object (all squares/rectangles) it would just change the slope to the opposite based on what side ...
0
votes
3answers
515 views

Formula for gallons in a trough

I have a trough which is a circular container. How do I determine how many gallons of water it takes to fill up the trough? I was thinking that we measure the height and the width but I think it's a ...
25
votes
12answers
21k 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 ...
11
votes
2answers
444 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 ...
13
votes
9answers
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 ...
6
votes
4answers
3k views

Can a circle truly exist?

Is a circle more impossible than any other geometrical shape? Is a circle is just an infinitely-sided equilateral parallelogram? Wikipedia says... A circle is a simple shape of Euclidean geometry ...
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 ...
23
votes
5answers
894 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 ...
20
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 ...
6
votes
7answers
207 views

area of figure in sector of intersecting circles

I need to find an area of blue part of figure APBC. I draw line segments between B and C, between C and A, and got equilateral triangle. I'm stuck here. Please help. Thanks. |AB| = a, P is midpoint ...
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 ...
7
votes
3answers
339 views

find area of dark part

let us consider following picture we have following informations.we have circular sector,central angle is $90$,and in this sector there is inscribed small circle ,which touches arcs of sectors ...
7
votes
2answers
3k views

Inscribed kissing circles in an equilateral triangle

Triangle is equilateral (AB=BC=CA), I need to find AB and R. Any hints? I was trying to make another triangle by connecting centers of small circles but didn't found anything
6
votes
3answers
2k views

Definite integral: $\displaystyle\int^{4}_0 (16-x^2)^{\frac{3}{2}} dx$

The following integral can be computed using the substitution $x = 4\sin\theta~$ and then proceeding with $dx = 4\cos\theta~ d\theta~$, and evaluating the integral of $\cos^4\theta~$: ...
6
votes
2answers
2k views

Prove that three points are enough to draw/define one and only one circle

Prove that three points are enough to draw/define one and only one circle, how would this be done?
5
votes
13answers
9k views

how to find center of an arc given start point, end point, radius, and arc direction?

Given an arbitrary arc, where you know the following values: start point (x0,y0), end point (x1,y1), radius (r) and arc direction (e.g. clockwise or counterclockwise from start to end), how can I ...
2
votes
2answers
110 views

circular reasoning in proving $\frac{\sin x}x\to1,x\to0$

The classic proof for $\frac{\sin x}x\to1,x\to0$ is using a squeezing theorem based on arguments about areas of circles. But as far as I know, all deduction of formula of circles' area is based on ...
10
votes
1answer
308 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 ...
8
votes
2answers
324 views

If $0$, $z_1$, $z_2$ and $z_3$ are concyclic, then $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear

If the complex numbers $0$, $z_1$, $z_2$ and $z_3$ are concyclic, prove that $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear. I really can't seem to get anywhere on this problem, ...
7
votes
3answers
20k 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 ...
6
votes
3answers
13k views

Proof of Angle in a Semi-Circle is 90 degrees

There is a well known theorem often stated as the angle in a semi-circle being 90 degrees. To be more accurate, any triangle with one of its sides being a diameter and all vertices on the circle has ...
5
votes
1answer
2k views

Closest point on circle edge from point outside/inside the circle

Alright, I am programming a plugin for a game that requires me to get the closest point on a circle when all you have is a point B, which is outside of the circle, the radius of the circle, and the ...