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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2answers
733 views

The area of circle

The question is to prove that area of a circle with radius $r$ is $\pi r^2$ using integral. I tried to write $$A=\int\limits_{-r}^{r}2\sqrt{r^2-x^2}\ dx$$ but I don't know what to do next.
2
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2answers
173 views

Calculate the closest point to the center of a circle from another circle on its radius.

There are 2 circles, the smaller one has its center on the bigger circles border, from that how can you calculate the coordinates the closest point on the smaller circle to the center of the bigger ...
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4answers
4k views

Area of a Circle Inscribed in a Square

A circle is inscribed in a square. The diameter of the circle is 12.4 mm. Find the area of the region that is outside of the circle and inside the square. Round the answer to the nearest tenth.
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0answers
53 views

Rotation of point with infinite child objects. (Chain rotation)

More of a thought experiment here, knowledge for knowledges sake. Let's say you can create infinite points that rotate smoothly and at the same speed as each other through a full revolution - let's ...
2
votes
0answers
115 views

Packing circles in circle vs semicircle vs quarter of circle

Consider $N$ disjoint circles with radius $1$ packed into a larger circle $C$. Let $R$ be the smallest possible radius of $C$, allowing the best packing density. Now take the $N$ unitary circles ...
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1answer
54 views

Radius of a circle

I'm having trouble with a question where I'm given two points, (-5,-2) & (1,0). Find the equation of the circle. I've used the midpoint formula to get the center which is (x+2) & (y+1) If I'm ...
1
vote
2answers
124 views

internal rectangle area intersected by a circle

I need to compute the internal rectangle area intersected by a circle like (the blue area) on these 3 examples: I know every vertex (x,y) coordinate and then their distance from circle center but ...
40
votes
17answers
5k 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 ...
2
votes
1answer
50 views

Calculating a specific point on a circle

I am looking for a formula to calculate the point of intersection where the arc crosses the angled line (designated by the letter 'X' in the example below), only from the dimensions given. I am ...
0
votes
1answer
45 views

Circles and tangents

3 circles of radius 3 cm, 4cm, 5 cm touch each other externally at $A$, $B$, $C$. Tangents drawn at $A$, $B$, $C$ intersect at $P$. Find $ PA + PB + PC$ . Thanks. My thoughts and approach: ...
1
vote
2answers
133 views

Proof when the circle map is ergodic

Let $E=[0,1)$ with Lebesgue measure. For $a \in E$ consider the mapping $\theta_a:E \rightarrow E, \ \ \theta_a(x) = (x+a) \mod \ 1$. a) Show that $\theta_a$ is not ergodic when $a$ is rational. ...
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1answer
236 views

What proportion of the circle is covered by the square?

Or what is the combined area of the circle segments (chords)? Picture a circle which is covered by a square, where the bottom vertices of the square are inscribed within the circle (so that the ...
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vote
2answers
49 views

Locating a point on a circle

I am having trouble getting the $(x,y)$ of a certain point on the circle. Please look at the image: The circles are the identical, the radius is $1000 \text{ units}$, $S$ is the center with ...
1
vote
1answer
44 views

Area of an ellipse proportional to integral of cross-ellipse distances?

I am curious if the area of an ellipse can be shown to be proportional to the integral of all cross-ellipse distances. Before I define cross-ellipse distance, I will give a motivating example from a ...
1
vote
2answers
68 views

How to find angular distance between points? [duplicate]

I have the following problem. I have several points on the plain, and there is another point somewhere in the middle of them. The goal is to find angular distance between any two points. My only ...
3
votes
3answers
899 views

How can I find the smallest enclosing circle for a rectangle?

I have the four vertices of a rectangle. I need to find it's smallest enclosing circle. For example: I need to find the radius of the circle.
1
vote
4answers
203 views

Showing that a circle is “tangent” to the $x$-axis if and only if $\left|k\right| = r$.

The problem is this: to show that a circle of radius $r$ and center $(h, k)$ intersects the $x$-axis at exactly one point if and only if $\left|k\right| = r$. Using geometrical intuition, this ...
5
votes
1answer
242 views

Find if a point lies in all given circles

I have a set of n given circles. I want to find that if there exists at least one point that lies in all of the given circles. Is there a method to do so? I can ...
0
votes
2answers
1k views

Prove that two circles are congruent if their radii are equal

Is this to be proved by showing that the circumferences/areas are equal?
2
votes
4answers
158 views

Find the equation of a circle which is tangent to $y$-axis at a given point and cuts a chord of given length on the $x$-axis

How to find the equation of the circle which touches $y$ axis at $(0,3)$ and cuts a chord of length $8$ on the $x$ axis? It should look like this: My approach: Since the circle touches $y$ ...
2
votes
2answers
73 views

Get the angle in a circle using center, radius and one point in a circle.

There is a circle and i know Point1 this is fixed and i know another point Point2 which can be anywhere in the circle. and i want to know the angle which is made at center. Thanks Your help will be ...
2
votes
3answers
853 views

Diameter of a circle using 3 nonlinear points

I am trying to find the diameter of a circle using 3 points on its circumference. 2 of the points are 5 feet from eachother while the third point is centered between the other 2. The ceter point is ...
0
votes
1answer
285 views

The surface area of a ring: $\pi[(r+dr)^2 - r^2]$ or $2\pi r\,dr$?

I know this may be really simple but here it is nonetheless. Let's say that I have a ring with a radius of $r$ and width of $dr$. I'm trying to find the surface $dS$ of the ring. Isn't it $dS = ...
5
votes
2answers
318 views

Rotation number of inverse maps on the circle.

I'm still a bit lost in my studies of rotation numbers. Any help is much appreciated! Let's say we have a homeomorphism $F: \mathbb{R} \rightarrow \mathbb{R}$ which is a lift of a homeomorphism ...
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2answers
113 views

Locus of vertex of triangle moving inside circle

A right triangle with sides $3,4$ and $5$ lies inside the circle $2x^2+2y^2=25$. The triangle is moved inside the circle in such a way that its hypotenuse always forms a chord of the circle. The locus ...
4
votes
4answers
910 views

True or False: The circumradius of a triangle is twice its inradius if and only if the triangle is equilateral.

Let $R$ be the circumradius and $r$ be the inradius. The if part is clear to me. For an equilateral triangle, the circumcentre, the incentre and the centroid are the same point. So, by property of ...
5
votes
7answers
840 views

Finding all the values of $\theta$ for which $\tan(\theta)=\sqrt3$; problem with understanding.

My textbook has a section where it says a possible way that $\tan(\theta)$ can be thought of is: For acute angles $\theta$, $\tan(\theta)$ is the $y$-coordinate of the point on the terminal side ...
2
votes
1answer
814 views

The locus of centre of circle tangent to two given circles

What is the locus of the centre of circles that are tangent to two given circles? I had no idea how to approach the problem so I considered a special case, namely one in which the two circles were ...
0
votes
1answer
660 views

finding points with maximum distance between them on a circle

I'm a computer science student working on a problem in computer graphics and looking for a formula that can find the x and y positions of a set of N points on the surface of a circle so that the ...
0
votes
3answers
153 views

Please find the radius of the circle.

Hello, I want to find the radius of red circle. I tried it with several ways like trigonometric. But there is a special value is given with this. I can not understand why it is given. It is the blue ...
2
votes
2answers
188 views

A question about 4 concyclic points

In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles ...
3
votes
1answer
218 views

Points of intersection between circle and parabola

Find the points of intersection between circle and a parable: circle: $x^2 + y^2 - 2x + 4y - 11 = 0$ parable: $y = (-x^2+ 2x + 1 - 2\sqrt{3})$ I don't understand how to solve this, I really tried, ...
0
votes
2answers
110 views

x(u,v), y(u,v), z(u,v) parametric equations for a special cycloid

I'm trying to find out a 3d parametric equations for a cycloid I know that a cycloid is a 2d curve it is generated by a point on a rolling circle. but my circle is rolling around another circle both ...
2
votes
2answers
62 views

If $ax + by = c$ is tangent to the circle $x^2 + y^2 = 16$

Problem: If $ax + by = c$ is tangent to the circle $x^2 + y^2 = 16$ then which of the following is correct option (A) $16 ( a^2 + b^2) = c ^2 $ (B) $16 ( a^2 - b^2) = c ^2 $ (C) $16 ( a^2 +b^2) = - ...
2
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1answer
113 views

From any arbitrary point $P$ on $y =\cos x$ tangents $PA$ and $PB$ are drawn to a circle which passes through

From any arbitrary point $P$ on $y =\cos x$ tangents $PA$ and $PB$ are drawn to a circle which passes through the points $(1,0)$ and $(3,0)$ and touches the circle $x^2+y^2-2x-8=0$ and have its ...
1
vote
1answer
55 views

Consider a series of n concentric circles $c_1,c_2 \cdots c_n$ with radii $r_1,r_2.\cdots r_n$ satisfying $r_1>r.$.

Consider a series of n concentric circles $c_1,c_2 \cdots c_n$ with radii $r_1,r_2.\cdots r_n$ satisfying $r_1>r_2>r_3 \cdots r_n$ and $r_1=10$ The circles are such that the chord of contact of ...
1
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0answers
86 views

The Biggest Smallest Piece to Smallest Biggest Piece ratio of a circle cut by n chords with maximal number of regions

It is well known that a circle cut by n chords gives at most (n^2 + n + 2 )/2 regions eg. http://mathworld.wolfram.com/CircleDivisionbyLines.html Questions:- How close to equal area regions can we ...
0
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0answers
164 views

Area covered by multiple (possibly intersecting) circles on surface of sphere

I have a number of circles of same radius on surface of sphere (Google Maps API). I'm trying to calculate the total area covered by these possibly intersecting circles. My current solution is ...
0
votes
2answers
421 views

Circle touching three tangential circles

The circles $C_1,C_2$ and $C_3$ with radii $1,2$ and $3$, respectively, touch each other externally. The centres of $C_1$ and $C_2$ lie on the $x$-axis, while $C_3$ touches them from the top. Find the ...
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2answers
92 views

Proving that four points are non-concyclic

In the figure, G and H are the circum-center and the orthocenter of ⊿ABC respectively. AH produced meets BC at O. GR ┴ BC at R. BS is the diameter of the circumscribed circle. Show that B, O, H, and ...
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2answers
852 views

what is the difference between tangent and slope of tangent?

I need your help, my question is what is the difference between tangent and slope of tangent ? A clear example would be appreciated. Thank you.
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2answers
78 views

Find $\angle BOD$ in the given figure.

Consider a circle with centre $O$. Two chords $AB$ and $CD$ extended intersect at a point $P$ outside the circle. If $\angle AOC=43^\circ$ and $\angle BPD=18^\circ$, then the value of $\angle BOD$ is ...
2
votes
2answers
165 views

Furthest distance between two circles in 3D

I have two circles in 3D specified by their centers, $c_1, c_2$, their radii, $r_1, r_2$, and the normals of the plane each circle is embedded in, $n_1, n_2$. Note that $n_1$ is not necessarily ...
1
vote
2answers
4k views

Calculating the arc length of a circle segment

I would like to calculate the arc length of a circle segment, i.e. I know the start coordinates (x/y) of the circle segment, the end coordinates (x/y) and the x and y distances from the starting point ...
4
votes
3answers
170 views

what does tangent mean?

I need your help, my question is what does tangent value mean and how can we benefit from it ? I know that $\tan(\theta) = \dfrac{\sin(\theta)}{\cos(\theta)}$, but what does that mean? Sorry I am ...
2
votes
0answers
69 views

To find a fifth degree equation by using circles and lines that cannot be solved by radicals

An example quintic whose roots cannot be expressed by radicals is $x^5 - x + 1 = 0$. I asked a geometry question about a fifth degree equation long time ago . I had an equation in the question. It ...
3
votes
1answer
451 views

How many points does 'the-most-point-contained-circle' contain at least?

Question : Given any $n$ distinct points $S$, consider the $\binom n2$ discs $D_{pq}$ formed by picking a pair of points $p,q$ from $S$ and using them as a diameter. For each disc $D_{pq}$, let ...
0
votes
3answers
51 views

Find the point in the circle

The circle $C$, given by the equation: $$x^2 + y^2 + (1+k)y - (k+1) = 0 $$ pass through the same two points for every real number $k$. Find the coordinates of these two points. Find the minimum ...
3
votes
4answers
166 views

A problem on circle

Consider a circle $C$, such that $\overline{AB}$ is a chord. $P$ be a moving point on the circumference of the circle. (i) How to find the point $P$ such that $\overline{PA}\cdot \overline{PB}$ is ...
2
votes
2answers
580 views

Circle areas on squared grid

There is a circle. On 9 equal squares. Every square has some value assigned to it. Every square gets weight, depending of what percentage of it is circle (area-wise). I need to find circle radius, ...