What is Circle: Definition and 1000 Discussions

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. A

    Number of ways to place n numbers in a circle?

    Homework Statement In a circle we can place k numbers. The numbers can range from 1 to n. One position in the circle is fixed, say by 1. We have to place the other k-1 places with numbers in the range 1 to n such that no adjacent numbers are equal. Homework Equations The Attempt at...
  2. B

    MHB Trig problem: max square cut from circle

    [SOLVED]Trig problem: max square cut from circle Since this one is homework related, ill ask this as a procession so that I can make sure that I am grasping the logic. the question is that with a round log with a diameter of 16cm, what is the strongest log that can be cut where the strength =...
  3. S

    Determining whether grid coordinates lie within a circle

    I have a grid and want to determine whether a point lies within (our outside of) a circle. The grid cells simply have integer coordinates, e.g. x = 5, y = 7. The circle's radius is known, and also an integer value. I wrote a program that can place points in a (quantized) circle using...
  4. S

    Nonconducting Rod bent into Circle

    Question A thin nonconducting rod with a uniform charge distribution of positive charge Q is bent into a circle of radius R. The central perpendicular axis through the ring is a z axis, with the origin at the center of the ring. What is the magnitude of the electric field due to the rod at...
  5. S

    Random Walk on a Circle: How Does the Last Unique Position Visited Distribute?

    Homework Statement Consider a random walk on a circle of N points, labeled {0,1,...,N-1}. Let the initial state be X = 0 and define T to be the first time all points have been visited at least once. Show that the distribution of X[T] (i.e. last unique position visited) is uniform over...
  6. icesalmon

    Calculating Arc Length of a Circle: What's the Correct Formula?

    Homework Statement Find the Arc Length from (0,3) clockwise to (2,sqrt(5)) along the circle defined by x2 + y2 = 9 Homework Equations Arc Length formula for integrals The Attempt at a Solution I have the correct answer at 3arcsin(2/3), but I tried to do this without calculus the first time...
  7. C

    Breaking static friction w/ a circle

    Homework Statement A crate (35 kg) is located in the middle of the flat bed of a pickup truck as the truck rounds an unbanked curve in the road. The curve may be regarded as an arc of a circle of radius 80.0 m. If the coefficient of static friction between crate and truck is 0.410, how fast...
  8. B

    Help drawing Mohr's circle with rotated axis

    Homework Statement Plane stress in xy-plane. Use Mohr's Circle to find σx1 σy1 and τx1y1 if the XY axis is rotated counterclockwise θº I want to do my HW problem myself so I'll just put some sample values. If i know how to get this one, i'll know how to do the HW problem(s). (Units won't...
  9. G

    Will a rectangular truck fit into a half circle formed tunnel when

    A truck with a width of 2,40 and height of 3,41 is driving through a tunnel that is formed like a half circle and has the radius of 3,60. Will it work? Now, this is a very pathetic question because this is junior high level geometry. I could blame the incapacity to solve this question on...
  10. C

    Tension, speed, in a circle

    Homework Statement A 68.8 g steel ball is attached to the end of a string. The ball is swung in a vertical circle 86 cm in diameter. (a) What is the minimum speed the ball must have at the top in order to complete the circle without falling? (b) If this speed is constant, what will be the...
  11. S

    Calculate the area of the circle

    i tried hard to solve this question but i got a complicated answer any hint ? http://www.gulfup.net/uploads/13634557771.gif thanks
  12. Saitama

    Finding radius of circleWhat is the formula for finding the radius of a circle?

    Homework Statement (see attachment 1) In this problem, we have a row of circles placed on a line. All points of tangency are distinct. The circle ##C_n## is uniquely determined. Homework Equations The Attempt at a Solution Here's the sketch I drew for the problem. Radius...
  13. stripes

    Uniform convergence for heat kernel on unit circle

    Homework Statement I would like to use the Weierstrass M-test to show that this family of functions/kernels is uniformly convergent for a seminar I must give tomorrow. H_{t} (x) = \sum ^{-\infty}_{\infty} e^{-4 \pi ^{2} n^{2} t} e^{2 \pi i n x} . Homework Equations The Attempt at a...
  14. E

    Shifting a 'complex' circle

    We typically have z=r*e^(i*theta). But let's say I want a circle centered at 2i. Is it valid to write z=r*e^(i*theta)+2i ? I ask this because I don't want to have abs(z-2i)=r; I want to solve for z.
  15. E

    Area of a circle by integration

    My curiosity was piqued by another poster who's trying to find the area of a lune using calculus. I wanted to do this now. So I don't highjack his thread, I'm making a new post about it. First, the geometric picture of a lune that constitutes my basic plan of attack. First I want to solve a...
  16. L

    When do SO(2) actions on the circle in the plane determine a metric?

    a metric on the plane determines an action of SO(2) on is unit circle by rotation. Suppose one starts with a free transitive action of SO(2) on a circle. When does this come from a metric? Always?
  17. S

    Mutual Inductance Between Square and Circle Circuit

    Homework Statement There are two circuits in the XY plane: one is a square (side 0.2 m in length) centered on the origin. The second is circle or radius r also centered on the origin. The circle is smaller than fits inside) the square. By assuming the radius of the circle is small compared...
  18. M

    MHB The Unit Circle, the Sinusoidal Curve, and the Slinky....

    I seem to recall when taking college Trigonometry my professor saying that the unit circle and sinusoidal curves were basically a mathematical represention of a slinky in that the unit circle was the view of a slinky head on, so that what you saw in the two dimensional sense was a circle, and...
  19. P

    Centre of a circle & complex numbers

    arg(\dfrac{z}{z-2}) = \dfrac{\pi}{3} sketch the locus of z and find the centre of the circle I've sketched the locus of z but I can't seem to find the centre of the circle. Is there a way to do it algebraically? I've attempted to use z = x + iy, but to no avail.
  20. S

    Center of Mass with Circle that has a Cut Out

    Homework Statement Find the center of mass of a 5kg circle with a radius of 4m with a circle that has a radius of 2m cut out from it. Homework Equations The Attempt at a Solution To be completely honest, I do not know where to start. I do know how to find the center of mass when...
  21. J

    Is a Circle Homeomorphic to a Subset of R^n?

    Hi, how can I prove that a circle it is not homeomorphic to a subset of R^n I can somehow, see that there isn't any homeomorphic application, for example between a circle in R^2 to a line, but how can I prove it? Thank you
  22. S

    Can Green's Theorem be used to evaluate line integrals over circles?

    Homework Statement Solve: ∫(-ydx+xdy)/(x2+y2) counterclockwise around x2+y2=4 Homework Equations Greens Theorem: ∫Pdx + Qdy = ∫∫(dQ/dx - dP/dy)dxdy The Attempt at a Solution Using Greens Theorem variables, I get that: P = -y/(x2+y2) and Q=x/(x2+y2) and thus dQ/dx =...
  23. S

    Lattice Points on a Circle

    Given a circle centered at the origin, how can one prove that the limit of the quotient of the number of lattice points on the circle over the radius goes to zero as radius goes to infinity?
  24. Y

    Polarization and Poincare circle.

    What is the theory behind mapping of the latitude and longitude of the sphere in the Poincare Circle to the polarization of the TEM wave? That is, why: 1) Linear polarization when ε=0 deg? 2) Circular polarization when ε=+/- 45 deg? 3) Elliptical when ε is not 0 or +/- 45 deg? 4) RH rotation...
  25. M

    Geometry: Triangle with a Circumscribed and Inscribed Circle

    Homework Statement What is the area of a right triangle whose inscribed circle has radius 3 and whose circumscribed circle has a radius 8? Homework Equations The diameter must be the hypotenuse of the circle The Attempt at a Solution The answer is 57, but I do not know the...
  26. A

    Bio-Savart Law, current through wire in semi circle

    Homework Statement I solved this problem. I just have a general question at the very end. A current flows in a wire that has straight sections on either side of a semicircular loop of radius b. Find the mag and direction of the magnetic field B at point P(center of loop). ......... periods =...
  27. Z

    Calculating a Circle Through 3 Points - Equations and Confusion

    Homework Statement I was looking at the following tutorial http://mathworld.wolfram.com/Circle.html Homework Equations equations 31-34 o the link The Attempt at a Solution My question is just whether this means that for 31-34, the answers are determinants of 3x3 matricies...
  28. L

    Find volume of circle. Cross-sections are squares. What am I doing wrong? :/

    Hi, everyone. I am just trying to do some practice problems on finding volume. So this is the one I'm working on: 1. "Find the volume of the solid described below: The solid lies between the planes perpendicular to the x-axis at x=6 and x=-6. The cross-sections perpendicular to the axis on...
  29. L

    Circle equation vs. semicircle equation

    This may sound like a dumb question, but... For the equation of a circle with a radius of 1, the equation is: x^2 + y^2 = 1 But if we rearrange that to be: y= √(1 - x^2), then it's only a semicircle... Why is that? Why is it now a semicircle just because it got rearranged? Thanks!
  30. T

    Maximise perimeter of triangle in a circle

    Hey guys, I hope someone can give me some pointers with this because it should be really easy but I am just not getting it! I want to show that for a triangle inscribedin a circle an equilateral traingle gives the maximal perimeter. I've tried a few things and just get bogged down in algebra...
  31. Z

    Dynamics of circular motion. Particle on the top of a circle

    Homework Statement A particle of mass M is on the top of a vertical circle without initial velocity. It starts to fall clockwise. Find the angle with respect to the origin, where the particle leaves the circle. Homework Equations v=ωXr The Attempt at a Solution I used two unitary...
  32. K

    Question about circle chord midpt locus

    Homework Statement A is the pt where the circle with wquation x^2+y^2=25 cuts the positive x-axis. Find the midpts of the chords of this circle that contain the pt A Homework Equations The Attempt at a Solution Since it is about the midpt of chords, I try to set up a equation...
  33. L

    A Question about circle bundles

    This question asks whether every circle bundle comes from a 2 plane bundle. Paracompact space please - preferably a manifold. By circle bundle I mean the usual thing, a fiber bundle with fiber, a circle, that is locally a product bundle. The transition functions lie in some group of...
  34. 7

    Finding X,Y points on a Circle

    I would like to calculate (X,Y) coords on a circle with a 10" radius. I have some idea on how this can work but I'm not real solid on it. Say the center is (0,10) and I'd like to solve for Y given an arbitrary X. How would I do this? Obviously (10,0) and (-10,0) and (0,-10) and I know that sin...
  35. S

    Finding the center of an n-gon (circle) based on angle and side-length

    I hope this is self-evident to someone, I'm struggling. I have a program that draws circles (n-gons really) of various sizes, but by translating-rotating-translating-rotating-..., not by x=sin/y=cos. That works as intended, but my wish is to offset the circle so that its center is (0,0) in...
  36. T

    A question on degrees of maps of the fundamental group of the unit circle

    Hello, I'm reading a textbook and in the textbook we are discussing the fundamental group of the unit circle and having some difficulty making out what a degree of a map is and why when there is a homotopy between two continuous maps f,g from S^{1} to S^{1} why the deg(f)=deg(g) We have...
  37. M

    Tangent to the centrum edge of a circle

    Homework Statement Hi i have a circle that is shown by (x-7)2+(y+1)2=20 i also have a line y=2x-5 and i have to explain why the line is a tangent to the edge of the circle i know that the circle has the centre in (7,1) and that the radius of it is 4,4 Homework Equations i know...
  38. F

    Radius of Convergence: Complex Series Need Not Be Defined Everywhere

    A complex series need not be defined for all z within the "circle of convergence"? The (complex) radius of convergence represents the radius of the circle (centered at the center of the series) in which a complex series converges. Also, a theorem states that a (termwise) differentiated...
  39. U

    Find the locus of center of circle

    Homework Statement A variable circle cuts x and y axes so that intercepts are of given length k1 and k2. Find the locus of center of circle Homework Equations The Attempt at a Solution There must be four intercepts but only two are given.
  40. Z

    Calculate double integral of the intersection of the ellipse and circle

    how to calculate the double integral of f(x,y) within the intersected area? f(x,y)=a0+a1y+a2x+a3xy The area is the intersection of an ellipse and a circle. Any help will be appreciated, I don't know how to do this. can I use x=racosθ,y=rbsinθ to transformer the ellipse and...
  41. A

    MHB Find the area of the four sectors of the given circle

    if we have the circle in the picture given x,y,z the middle line pass through the circle center find the area of the four sectors with respect to x,y,z parallel lines Thanks
  42. Q

    Four complex points lie on a circle proof

    This isn't really homework. I'm studying What Is Mathematics by myself. But I'm very stuck on one of its exercises. Homework Statement Prove that if for four complex numbers z_{1}, z_{2}, z_{3} and z_{4} the angles of \frac{z_{3} - z_{1}}{z_{3} - z_{2}} and \frac{z_{4} - z_{1}}{z_{4} -...
  43. C

    Finding Points of Tangency for the Unit Circle

    Hi I'm trying to study over break, this isn't an exact quote but its the part of the problem I need help with. Thanks. Homework Statement Draw the unit circle and plot the point P=(3,2). Observe there are TWO lines tangent to the circle passing through the point P. Lines L1 and L2 are...
  44. D

    Unit circle and the roots

    Homework Statement Calculate the integral ∫dθ/(1+acos(θ)) from 0 to 2∏ using residues. Homework Equations Res\underline{zo}(z)=lim\underline{z->zo} (z-z0)f(zo)*2∏i The Attempt at a Solution To start I sub cos(θ)=1/2(e^(iθ)+e^(-iθ)) so that de^(iθ)=ie^(iθ)dθ Re-writing in...
  45. O

    Car speeding up around a circle

    Homework Statement A car is being driven around in circles. The radius of the circle being made is R = 150.0 m. At t = 0, the car is on the left edge of the circle (therefore it is in the −x direction away from the center of the circle if your origin is placed at the center), and it is...
  46. D

    MHB How do I add directional arrows going counterclockwise on the circle?

    \begin{tikzpicture} \draw (0,0) circle (1cm); \end{tikzpicture} How do I add directional arrows going counterclockwise on the circle?
  47. P

    Definite integral of a quarter circle

    Homework Statement Find the definite integral of a quarter circle. Homework Equations x^2 +y^2=10 The Attempt at a Solution x^2=10-y^2 x=sqrt(10-y^2) ∫ sqrt(10-y^2)dy from 0 to sqrt(10) I'm not sure what to do here.
  48. A

    MHB Equation of a line tangent to a circle

    The circle x^2 +y^2 -4x+2y+m=0 is tangent with the line y=x+1.Find m. p.s : I know that o should solve it from the equations of two lines but i really get confused when i substitute the y :/ . Thanx :)
  49. B

    Segment of a circle calculation

    I'm working to calculate the cross-sectional area of a lathe turned feature machined with a radiused insert. My calculations have essentially led me to the equation for the area of a segment of a circle. Area=r^2/2*(∏/180*C-sin(C)) where r is the circle's radius and C is the central angle...
  50. 7

    LaTeX Fill with color three of five sectors in circle with latex.

    Hi every one, here is a question about drawing and filling in Latex (\usepackage{tikz}) I just draw a circle with 5 sectors correctly, and what I need help on please, how I fill 3 sectors with green color and the other 2 sectors with red... please help me with this task...
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