Circle Definition and 1000 Threads

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.
Specifically, a circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a disc.
A circle may also be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations.

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  1. M

    How to Construct a 1 Degree Arc Using a Given 19 Degree Arc?

    Exercise #27 from a textbook called Kiselev's Geometry / Book I. Planimetry: Using only compass, construct a 1 degree arc on a circle, if a 19 degree arc of this circle is given.Please, check my reasoning on this one. I just want to make sure that I'm getting it right. My solution: Using a...
  2. C

    MHB Determining Diameter of Pipe with Drilled Hole

    A textbook shows a circle with a line drawn through the centre and advises that the line represents the diameter of the circle, ok with that, but what happens if the circle was a pipe and the pipe had a hole drilled into it, so the pipe now has an outer diameter of 10 but the hole inside the...
  3. J

    Geometry question with the perimeter of a circle

    say I have a circle with a diameter of d and there is a point at the top of the circle, p. I want to know the distance from point p to any other point on the perimeter and the angle θ from the tangent line of p. is there a function that will describe this? I will try to put up a picture to...
  4. M

    Diameter of circle of light seen from above water - illuminated beneath

    Homework Statement A small light is 22.0 cm below the surface of a liquid of refractive index 1.50. Viewed from above, the light appears to illuminate a circle on the surface of the water. What is the diameter of the circle? cm Homework Equations Snell's Law n1sin(θ)1 = n2sin(θ)2...
  5. I

    Uniform Distribution on unit Circle

    I keep reading that a random vector (X, Y) uniformly distributed over the unit circle has probability density \frac{1}{\pi}. The only proof I've seen is that f_{X,Y}(x,y) = \begin{cases} c, &\text{if }x^2 + y^2 \leq 1 \\ 0 &\text{otherwise}\end{cases} And then you solve for c by integrating...
  6. K

    How does one get the form of a circle out of this equation?

    Homework Statement Sketch modulus((z+1)/(2z+3))=1 on the complex plane where z=x+iy Homework Equations The Attempt at a Solution I know it is a circle but i need help simplifying the equation into the form of a circle. i'm stuck at 0= 3x^2 + 3y^2 + 10x + 8 I usually...
  7. M

    Solving a equation of a circle

    Find the center and radius of the circle using the equation: x^2 + y^2 + 2√2x - 4√5y = 5 I just can't seem to solve this equation into the form (x-h)^2 + (y-k)^2 = r^2 in order to get the center and radius. Any help would be appreciated
  8. D

    Centroid of a TILTED Semi circle

    Centroid of a "TILTED" Semi circle! Guys, could anyone help me out how to solve the centroid of these "tilted" semicircle. I really don't have any idea how to get its centroid since it's tilted..Please I really want to understand so kindly do it step by step, Thanks.. Just Find the x-bar and...
  9. H

    Calculating angle of circle to produce given ellipse

    I know I should keeps this short, but I need to explain it a little. So, please have patience with me :) Given a standard ellipse, and its eccentricity and position on screen, is there some clever way of calculating how much a regular circle needs to be tilted in (angles) in a perspective...
  10. M

    MATLAB Plot unit circle in chebychev metric in MATLAB

    Ok, so I'm trying to plot the unit circle using the chebyvhev metric, which should give me a square. I am trying this in MATLAB, using the 'pdist' and 'cmdscale' functions. My uber-complex code is the following: clc;clf;clear all; boundaryPlot=1.5; % Euclidean unit circle for i=1:360...
  11. X

    Equation of a Circle with a Center and Tangent Point

    What is the equation of the circle with a center point of (10, -14) when the circle is tangent to x=13? D = √(13-10)^2 + (0-(14))^2 D = √(3)^2 + (14))^2 D = √9+196 D = √205 Radius = √205 (x-10)^2 + (y-(-14))^2 = √205^2 (x-10)^2 + (y+14)^2 = 205 But how am I suppose to graph this?
  12. S

    Trying to find an angle in lines layed on top of a circle

    Homework Statement The problem is attached as TheProblem.jpg and the answer is A. Homework Equations Geometry rules. The Attempt at a Solution The triangle which has angles of 30deg and 10deg also has an angle of 180deg-30deg-10deg = 140deg and the other side of the line intersection...
  13. S

    Mapping unit circle from one complex plane to another

    I want to show that if the complex variables ζ and z and related via the relation z = (2/ζ) + ζ then the unit circle mod(ζ) = 1 in the ζ plane maps to an ellipse in the z-plane. Then if I write z as x + iy, what is the equation for this ellipse in terms of x and y? Any help would be...
  14. G

    Calculating the Area of a Circle in the First Quadrant | Integration Homework

    Homework Statement -8\int^{3}_{0}\sqrt{9-x^2}dxHomework Equations The Attempt at a Solution am i right in thinking this the area of a circle in the first quadrant so my answer is-8(\frac{9\pi}{4}) = -18\pi Thanks for reading?
  15. S

    Simple linear algebra problem (points on circle for a given vector and angle)

    Hi all, I have a seemingly simple linear algebra problem which I have trouble with and I would like to ask for some advice how to solve it. Here is the problem: Here are my thoughts about this: It is clear that a solution is not always defined for the whole range of \nu and that for...
  16. S

    Finding the Mass of a Circle Using Double Integration?

    Homework Statement I would like to ask how to find the mass of a circle with equation x^2+y^2=4 given its density=xy^2 by not using polar coordinate but use dxdy or dydx ( cut the circle into pieces parallel to x-axis or y-axis ) Homework Equations x^2+y^2=4 xy^2 The...
  17. F

    Which is Larger: Circumference of an Inscribed Circle or Triangle Perimeter?

    Homework Statement A circle is inscribed in a triangleHere is a picture Picture of circle inscribed in triangle, not necessarily to scaleWhich is larger: the circumference of the circle, or the perimeter of the triangle? Homework EquationsC=∏D (D=diameter of the circle, C=circumference of...
  18. Feodalherren

    Maximize area of rectangle inside a circle

    Homework Statement Show that the maximum possible area for a rectangle inscribed in a circle is 2r^2 where r is the radius of the circle.Homework Equations Pre-calc ! NO TRIG ! Doesn't matter if it's easier, it's supposed to be solved with algebra. The Attempt at a Solution I have no clue...
  19. C

    Acceleration of a 2D circle due to Gravity.

    Hello Im wondering how to calculate the acceleration of a circle down an inclined plane (due to gravity). I am familiar with caclulating the acceleration of a body sliding down a inclined plane, but not a circle. How do you determine the acceleration of a circle (preffer rotation per second, if...
  20. V

    Analytic Geometry Question (equation of a circle)

    Homework Statement Find the equation of the circle whose centre lies on the x-axis and which passes through points A (6,0) and B (0,10). Homework Equations The Attempt at a Solution I drew a diagram of the circle and determined that the line AB has gradient 5/3. Its perpendicular bisector...
  21. Feodalherren

    Triangle inside circle, find area of circle.

    Homework Statement An equilateral triangle of side x is is inscribed in a circle. Express the area of the circle as a function of x. Homework Equations Anything non-trig. I suspect it's got something to do with the Pythagorean theorem. The Attempt at a Solution I tried getting the...
  22. M

    Circle & Ellipse Intersection: Can you Make Them Touch?

    What does it mean when one says that "A circle and an ellipse with a focus at the circle’s center can touch each other only at the longer axis"? Can't you, by varying the size of the circle, make it intersect the ellipse in a variety of ways? Thanks! :)
  23. O

    Question relating to shifting a circle.

    This problem arose for me while working out a triple integral in spherical coordinates. Basicly I know that when we shift a parabola along the axis it is simply translated. I naturally assumed that if we shifted a circle in a similar manner that it would act the same. However when we shift...
  24. G

    Lines on a circle - a counting problem

    So over the course of yesterday and the day before that I've spent a few hours thinking about these problems; 1. Given a circle, place n points arbitrarily* on the edge and connect every point to every other point with a straight line. How many intersections do you get? 2. Given the same...
  25. Z

    Confusion of circle and sphere for physics problems

    Homework Statement The problem is attached Homework Equations P=F/A The Attempt at a Solution I did the question like this (got wrong answer though): Surface area of sphere=4∏r2=4×∏×0.252 Atmospheric pressure=1.01×105 Force=1.01×105×4×∏×0.252≈80000N Actual answer 20000N...
  26. K

    What is the area of this part of a circle?

    Homework Statement the figure is shown on the attachment. Find the area of the smallest part of the circle. Homework Equations area of circle, sector, segment The Attempt at a Solution I cannot use the said equations since the part of the circle is not with reference to the...
  27. S

    Center of mass of half square without a half circle

    Homework Statement Find the position of the center of mass for a thin sheet and homogeneous, with sides R and 2R ,from which has been subtracted a half circle of radius R. [Xcm=(2/3)*R*(4-pi)]Homework Equations Rcm=(1/M)*∫rdm The Attempt at a Solution By symmetry we know Ycm=0. For de...
  28. zoobyshoe

    News Assad's inner circle trying to covertly defect to rebels

    I thought this was a very interesting development in the Syrian situation: http://www.jpost.com/MiddleEast/Article.aspx?id=274828
  29. M

    Conic Sections on the Complex Plane (circle)

    Homework Statement Describe the locus and determine the Cartesian Equation of: \left|z-3-5i\right|= 2 Homework Equations \left|z-C\right|= r -----> formula for a circle on complex plane Where C = the centre z = the moving point (locus) (x-h)^{2}+(y-k)^{2}=r^{2} -----> Formula...
  30. estro

    What is the relationship between the integral and the area of half a circle?

    \int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}} \sqrt{r^2-x^2-y^2}dy I can calculate the above integral [part of a double integral] by the conventional way [somewhat long], however my book says that this integral equals to \frac {\pi}{2}(r^2-x^2) because the integral is actually the area of half a...
  31. M

    MHB Finding the Circumference of a Circle Using Improper Integrals

    Hello again, I'm finding myself stuck on what is probably a simple question, but I believe I am taking the wrong approach. The section is "Volumes with infinite integrals," and the chapter is "Improper Integrals." The question, "Use calculus to find the circumference of a circle with radius...
  32. T

    Prove the ratio of AX:XB = 1:λ if X is a point on a circle

    Homework Statement The points P and Q divide a given interval AB internally and externally respectively in the ratio 1:λ. The point X lies on the circle with diameter PQ. Prove that AX:XB=1:λ Homework Equations None The Attempt at a Solution Basically, if we define the centre of PQ and the...
  33. W

    Radius of Inscribed Circle in a Quadrant of a Circle

    Homework Statement Find the radius of a circle inscribed in a quadrant of a circle with radius 5 Homework Equations The Attempt at a Solution I worked this but I'm not sure if its correct. I looked at the first quadrant so a quarter of a circle with radius 5. I drew the radius...
  34. D

    How to calculate dA of a circle

    Homework Statement Hello! I need to calculate a little fraction (dA) of the area of a circle (mass m and radius R and area A) and I have no idea how to do this. Homework Equations According to my textbook dm=\frac{M.dA}{A}=\sigma .dA and dA=R.dR.d\theta. The Attempt at a Solution Well, I tried...
  35. G

    Probability that a Rectangle lies within a circle

    This is not really a homework problem, I'm just doing it as an exercise puzzle. I think I'm on the right track, but at this point I feel a little exhausted and would love a hint. Homework Statement Let C be a unit circle: x^2+y^2=1 . Let "p" be a point on the circumference and "q" be a point...
  36. J

    Understanding Vertical Circle Motion and Tension in Circular Motion

    This question is about circular motion in a vertical circle Question 1: would I be correct in assuming that the magnitude for tension is dependant on (a) the weight of the object (b) the position of the object with respect to the horizontal diameter of the circle So above the...
  37. C

    Prove Cone over Unit Circle Homeomorphic to Closed Unit Disc

    Homework Statement This question comes out of "Introduction to Topology" by Mendelson, from the section on Identification Topologies. Let D be the closed unit disc in R^2, so that the boundary, S, is the unit circle. Let C=S\times [0,1], and A=S \times \{1\} \subset C. Prove that...
  38. M

    Is Work Done Always Zero in a Conservative Field?

    imgur.com/kBTVm Hi, I understand that work done in a conservative field when a closed loop is followed is zero. The answer to this question I am sure is B. How do I explain to my colllegue that it is not zero when he thinks that the speed is constant and there is no friction and the force...
  39. J

    Work done on body moving in a circle

    Situation 1: completely horizontal circle Imagine a ball being whirled in a completely horizontal circle. The only forces acting are tension and weight Would I be correct in assuming that in this situation NO WORK is done on the ball because the forces are directed (at all times)...
  40. B

    A particle in a 2d circle with potential

    Hello, What would be the right approach to solve for a particle's wavefunction/ energy eigenvalues inside of a 2d cicrle with a potential V(r) where r is the radial distance of a particle from the center of the circle? V(r) is known and is some sort of a well potential going to infinity at R...
  41. H

    Is a circle still considered a surface?

    The question asks to look for a surface and a circle is the only function which meets the conditions. Is this still considered a surface?
  42. L

    Dimension of the circle in the plane is 1

    I am becoming confused when I read in Wiki that the dimension of the circle in the plane is 1! It is said that the dimension of circle is 2 (in general )! I do not get it!
  43. D

    Circular motion- how slowly would you twirl the ball in vertical circle

    Homework Statement If you twirl a ball attached to a cord in a vertical circle, there would be a critical speed at the top for which the tension in the cord is zero. This is because the force of gravity supplies all the centripetal force necessary to complete the circle. How slowly would you...
  44. 1

    Circle radius 0, algebraic manipulation

    Homework Statement Going over some old tests, I am asked to find the contour of the function: T = 100 - x^2 - y^2 at T = 100, T = 0, etc. I have a question regarding the contour at T = 100 Homework Equations The Attempt at a Solution Consider T = 100 100 = 100 - x^2...
  45. N

    Analyzing Motion in a Circle: Acceleration & Velocity

    I was thinking about how planets revolve around sun. Although they subscribe a elliptical motion, my question is very similar. A heavy body exerts a force on a point mass, say with an acceleration of "a". If we take the direction of this acceleration to be X, what is the linear uniform velocity...
  46. O

    How to predict the shape of the circle from any point of view

    As we know a circle view at an angle appears as an ellipse , as you see in the picture, the center of the camera aim to the center of the circle , the angle between the circle axis and the camera is ө, the azimuth between mojor axis(a) and the camera is ∞, the rotation of the camera is €...
  47. A

    Finding equation of a circle given circumference and containing points.

    Homework Statement Find the equation of a circle if the circumference is 18∏ and contains the point (2, 8) The Attempt at a Solution I know I can find the radius by setting 18∏=2∏r. r=9. the equation of a circle is (x-h)2+(y-k)2=r2 So I have 92= (2-h)2+(8-k)2 which becomes...
  48. O

    How to predict the shape of the circle from any point of view?

    As we know a circle view at an angle appears as an ellipse, as you see in the picture, the center of the camera aim to the center of the circle , the angle between the circle axis and the camera is ө, the azimuth between mojor axis(a) and the camera is ∞, the rotation of the camera is €...
  49. V

    Parameterization of the Circle

    Homework Statement Consider the following parameterization of the circle: a) x1 (t) = (cost, sint) b) x2 (t) = (cos3t, sin3t) c) x3 (t) = (sint, cost) How long does it take point a particle to go from (1,0) to (0,1) for each parameterization. Homework Equations The...
  50. D

    Hit A Snag When Finding Area Inside A Circle And Under a Line

    Homework Statement Find the total area inside the circle r = 4 and below the line r=2csc\theta Homework Equations \int^{b}_{a} 1/2r^{2}\thetad\thetaThe Attempt at a Solution r=2/sin\theta\Rightarrowrsin\theta=2\Rightarrowy=2 r=4\Rightarrow=circle with radius 4 at center (0,0) Point of...
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