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. C

    Why the circle can't be homeomorphic to a real interval

    Hi guys! Excuse the spam, but I would like to ask something which I read in Armstrongs Basic Topology which I am just not 100% sure about. He says we wish to define homeomorphism such that a circle cannot be homeomorphic to an interval such as [0,1). A continuous function f : X \mapsto Y is...
  2. B

    How Can I Find the Centre Point of a Circle with Given Coordinates and Radius?

    given the x,y coordinates of two points and the radius of a curve. has anyone got a formula for calculating the coordinates of the centre point please
  3. K

    Work on Block in Vertical Circle: MasteringPhysics

    Homework Statement A small block with mass 0.0325 kg slides in a vertical circle of radius 0.475 m on the inside of a circular track. During one of the revolutions of the block, when the block is at the bottom of its path, point A, the magnitude of the normal force exerted on the block by...
  4. J

    Mass attached to a spring rotating in a circle

    Homework Statement A mass of 2kg rotates at 1m/s in a horizontal circle on a table at the end of a spring with an elastic constant of 50N/m. If the original length of the spring is 2m, find the extension of the spring. Given - M=2kg, V=1m/s, k=50N/m, original length of x=2m find x'...
  5. D

    Angular speed; horizontal circle

    Homework Statement A student ties a 400g rock to a 1.0m long string and swings it around her head in a horizontal circle. What angular speed does the string tilt down at a 10° angle? m = 400g = .4kg r = 1m Homework Equations F = \frac {mv^2}{r} \omega = \frac {|v|sin \theta}{|r|}...
  6. Jadaav

    Solve Circle Puzzle: 18 Math, 19 History, 16 Art

    I know, that probably this will sound to be a dumb question but I can't find the solution to it. So if anyone could help please ? Homework Statement In a class of 40 pupils, 18 take Mathematics 19 take History 16 take Art 6 take both Mathematics and History 5 take both History and Art 7 take...
  7. T

    Convergence on the unit circle

    Homework Statement Determine the behavior of convergence on the unit circle, ie |z| = 1 of: Ʃ \frac{z^{n}}{n^{2}(1 - z^{n})} Homework Equations Obviously this is divergent then z is a root of unity. The question is what happens when z is not a root of unity. The Attempt at a...
  8. S

    Fastest way to determine if a circle fully covers a rectangle

    As usual I'm working on a program and I'm having trouble with math/efficiency. Homework Statement I need a way to find out if a circle given as a point and a radius C(x,y,r) fully encloses a rectangle given by the top left corner and the width and height R(x,y,w,h) I only need to know...
  9. L

    Static friction on a car on a circle

    When a car is moving on a circular track which is sloped away from the centre of the track, is the static friction keep that car on the circle (constant speed) towards the outside of the circle, opposite the acceleration? I'm having trouble seeing this because to me, the car would want to...
  10. K

    Plot the sequence on the unit circle.

    Consider the sequence (n) n=1 to infinity. Plot the sequence on the unit circle: n modulo 2*pi for n≥1. What do you see? Attempt: I really honestly have no idea what to do. We are learning in class about limit laws and how to prove them, so this question seems to be coming out of nowhere. :(
  11. M

    Finding circle center from two points and an arc length

    I'm trying to find the equation for a circle given two points in x, y and the starting angle, arc length, and two points along the circle. I need to find the equation because I need to translate a sprite along the curved path from one point to another. The situation ends up looking like this:A...
  12. E

    Vertical Circle Motion: Acceleration at Top vs. Bottom

    Homework Statement A ball of mass m is suspended from a rope of length R. The ball is set into freely swinging circular motion a vertical plane. The centripetal acceleration of the ball at the top of the circle is 13g. What is the centripetal acceleration of the ball at the bottom of the...
  13. G

    Particle on top of a half circle

    Homework Statement A point-particle sits at rest at the top of a half-circle with radius R. Find the minimum initial velocity v0 the particle has to have in order to clear the half circle without rolling down on one of its sides. Homework Equations None are given. The Attempt at a...
  14. T

    2D Circle and rectangle intersection tests

    What I'm looking for is an algorithm to find the details on the intersection of of a circle and rectangle in two dimensional Euclidean space. The information I need to find is straightforward enough; all I need is to know whether the rectangle and circle are not intersecting, partially...
  15. J

    Statistics - arrangement in a circle

    Homework Statement Delegates from the G8 are to be seated around a circular table. How many different seating arrangements are possible if the French and Canadian delegates are to be seated next to each other, but the Russian and Japanese are not to be next to each other? Homework...
  16. B

    Intersection of a line and circle

    Homework Statement The line x+y=2a-1 intersects the circle x2+y2=a2+3a-3 at point (m,n). When m*n reaches its minimum value, what is the value of a?Homework Equations Equation One x + y = 2a - 1 Equation Two x2 + y 2 = a 2 + 3a - 3
  17. R

    Parameterization of a Circle Question

    This isn't really a HW question, it's just something that's been confusing me in my Calc class. We recently went over how to find curvatures of curves in 3D space. In lecture, the professor went over a simple example: a circle of radius 3 at any given point. Maybe it's because I don't...
  18. I

    Circle vs. Square edge problem

    I got a problem that I need to figure out. I have a piece of pipe 20 inches long and I bend it on a radius at 90 degrees. I measure from the back of the pipe to the end of one side and get 9, I flip it the other way and do the same thing and get 13.555. If you add those you end up with a...
  19. N

    Why is the formula for the surface area of a circle not (2*pi*r)^2?

    Actually I changed my mind and feel like it should be ((pi*r)(2*pi*r)) by my faulty thinking. Since pi*r would give you a line wrapped halfway around a sphere, I was thinking you could repeat this line in a radial pattern around the outside of a sphere (2*pi*r) times to get the surface area...
  20. B

    Find the coordinates of a point on a circle without knowing the center point.

    Given 2 points on a circle, call them A and B. I know the cartesian coordinates of A. I also know the radius of the circle, the slope of the tangent line at A, and the length and direction of the arc between A and B. I don't know the coordiates of the center of the circle. How do I find...
  21. C

    Approximate uncertainty in area of circle

    Homework Statement What is the approximate uncertainty in the area of a circle of radius 5.3 * 104 cm? Express your answer using one significant figure. Homework Equations A = pi*r2 The Attempt at a Solution Using the given radius, I found the area to be 8.8 * 109 cm2. And since...
  22. B

    Genus-g Surface and Retraction to Circle

    Hi, All: I saw an argument in another site re the claim that the genus-g surface Sg does not retract to a circle. The argument was that,using/assuming H_1(Sg,Z)=Z^{2g}; and H_1(C,Z)=Z ; Z the integers and H_1(Sg,Z) if there was a retraction r: Sg-->C , then , for i being the inclusion ...
  23. 5

    Wire cut to shape circle and square so total area is maximum or minimum

    Homework Statement A piece of wire 40 cm long is cut into two pieces. One piece is bent into the shape of a square and the other is bent into the shape of a circle. How should the wire be cut so that the total area enclosed is a (a) maximum and (b) minimum? Homework Equations Area...
  24. R

    My satellite should go in a circle, not an ellipse

    I have been toying with a java programming project for a few months now. I want to depict a satelite orbiting a planet (2 dimensional). I've scaled down certain constants to fit the screen and have put GM (mu) at 200000 and the distance from the gravitaional body, a planet, at 200...
  25. D

    Which trolley would turn in the smallest circle?

    Hi, my first post here is a simple one (hopefully) and is a from a series of mechanical comprehension questions I had been given. I have no idea which one is correct. If you do know the answer would you be able to give a brief explanation to why it is? Also, if anyone has any mechanical...
  26. E

    Why Does the Osculating Circle Formula Involve Derivatives?

    Hi guys! I learned yesterday what an osculating circle is and I am learning how to find the radius of curvature of some curves. For example I have found that for y=x^2 the radius of the osculating circle for the point [0,0] is 0.5 (That's why circular mirror works similarly to parabolic mirror...
  27. S

    Continuous formula for area of segment of a circle

    I'm writing a little program for generating some images, and at one point I need to calculate how much of a circle is on either side of a straight line that bisects the circle. The line is always vertical so it is easy to get the value of how much of a horizontal line segment within the circle...
  28. T

    Complex numbers on unit circle

    Homework Statement Let z1; z2; z3; z4 be four complex numbers on the unit circle (i.e., |z1|=|z2|=|z3|=|z4|=1). It is known that z1+z2+z3+z4=1+i . Find the value of 1/z1 + 1/z2 + 1/z3 + 1/z4 Homework Equations 1/z = barZ/|z|^2 The Attempt at a Solution I've been trying for about a day now...
  29. L

    Double Integral of a Circle with Limits of Integration

    Homework Statement Evaluate f(x,y)=y2\sqrt{1-x2} over the region x2+y2< 1 Homework Equations The Attempt at a Solution using x limits between -1 & 1 followed by the y limits of 0 & \sqrt{1-x2} \int\inty2\sqrt{1-x2}.dy.dx Evaluating this and multiplying be 2 to get the...
  30. 1

    Points on the outer edge of a circle

    My friend was telling me about something he read about involving points on a circle that seemed kind of cool. It went something like this: Have a circle of radius 1, and mark n equally spaced points on the outer perimeter. Choose one of the points, and connect all other points to it...
  31. A

    Area in cardioid and outside circle - Using Double Integral

    Area in cardioid and outside circle -- Using Double Integral Homework Statement Find the area inside of the cardioid given by r = 1 + cos\theta and outside of the circle given by r = 3cos\theta. Homework Equations \int\intf(x,y)dA = \int\intf(r,\theta)rdrd\theta not really relevant...
  32. Evil Bunny

    Electrical circuits don't flow in a circle?

    So I came across http://amasci.com/miscon/eleca.html#circle" on W Beaty's site and it's not sitting well with me. Here is a quote: This doesn't seem right to me. My impression has always been that the charges flow from one of the slots (hot) in the wall socket and return on the other...
  33. R

    Area of circle inscribed with 3 smaller circles

    Homework Statement A large circle is inscribed with 3 smaller circles, eachhttps://www.physicsforums.com/newthread.php?do=newthread&f=156 of the four circles is tangent to the other three. If the radius of each of the smaller circles is a, find the area of the largest circle. Homework...
  34. R

    Who is right, me or my professor? proof area of a circle

    Homework Statement "Set up an integral involving a function and evaluate the integral to prove the formula for the area of a circle of radius r is pi*r^2. Show all steps." 2. The attempt at a solution I imagined the circle as an infinite tiny arc lengths or "circumference's", with each arc...
  35. D

    Equation for lines that are tangents to a circle

    Homework Statement Find a differential equation whose solution is a family of straight lines that are tangents to the circle x^2+y^2=a^2 where a is a constant. The Attempt at a Solution So actually I'm stuck on the first part, coming up with such an equation. After some work I came up with...
  36. 2

    Determine angle of intersecting lines inside a circle

    So I ran across this problem on the 'net and I can't determine "x". The arc length of the circle is 360. I added some other variable and took what I know about a circle and intersecting lines. I wound up with four variables and four equations. x = 1/2 (y + 67) w = 1/2 (z + 147) y +...
  37. G

    Solving time for a satellite to circle the earth using velocity equation

    Homework Statement The average speed of an orbiting space shuttle is 19800 mi/h. The shuttle is orbiting about 233 mi above the Earth’s surface. Assume the Earth’s radius is 3963 mi. How long does it take to circle the earth? Answer in units of h. Homework Equations I think it...
  38. K

    Moment of inertia of a wire shaped into a semi circle

    Find the moment of inertia of a wire, AB, of mass M and length pi*a, which is bent into a semicircle, about AB. Mr^2/b] [b]The mass of the wire is M=pi*a*m, where m is the mass per unit length of the rod. Then a small element, ds is regarded, of the circumference of the semicircle as being...
  39. R

    Optimizing Pendulum Release Height for Maximum Tension and Circular Motion

    The pendulum bob in the above figure must circle the rod interrupting its swing, and the string must remain taut at the top of the swing. How far up must the bob be raised before releasing it to accomplish these goals? I don't know where to begin with this because I don't quite understand...
  40. D

    Offset Circle Equation: Determining Radius from Angle of Rotation

    Hi, I’m trying to write a probing cycle on a CNC for calibrating from a standard. I have a circle with a known diameter but it is not located on the center of rotation – the center of rotation is X/Y intercept. The center of rotation will always lie within the circle. I’m trying to...
  41. M

    Does a circle have a constant rate of change of it self that defines

    Does a circle have a constant rate of change of it self that defines it as a circle. I have never heard of such a thing but I am curious. It must right?? Since every circle is the same no matter the radius(the arch will change by some constant amount per radian) If that doesn't make...
  42. D

    Radius of a circle that intersects two points on a right triangle.

    Homework Statement I'm trying to figure out the radius of a circle that intersects two points on a right triangle. One side of the triangle is tangent to the circle and the other intersects it. I have attached an image that helps further explain what I'm talking about. Knowing what I have...
  43. D

    Radius of a circle that intersects two points on a right triangle.

    I'm trying to figure out the radius of a circle that intersects two points on a right triangle. One side of the triangle is tangent to the circle and the other intersects it. I have attached an image that helps further explain what I'm talking about. Knowing what I have listed in the image is...
  44. S

    Can say me why annulus and circle are not homeomorphic?

    Can say me why annulus and circle are not homeomorphic?
  45. P

    Understanding the Unit Circle and Trigonometry Functions

    I am learning about the unit circle and I am a bit confused. So, I have my circle drawn with radius 1 and I sketched a right angled triangle inside it so that the hypothenuse has a length of 1. I think what is making me confused is the meaning of sine, cosine and tangent. They are...
  46. I

    Proving Shortest Path b/w Two Points on a Sphere is a Great Circle

    Homework Statement proof that shortest path between two points on a sphere is a great circle. Homework Equations Euler-Lagrange and variational calculus The Attempt at a Solution in sphereical coords: N.B. \dot{\phi} = \frac{d\phi}{d\theta} ds = \sqrt{r^{2}d\theta^{2} +r^{2}sin^{2}\theta...
  47. Femme_physics

    Equation for a circle plugging for x and y, not getting a

    Homework Statement http://img40.imageshack.us/img40/7174/thecircleu.jpg A circle whose equation is http://img830.imageshack.us/img830/5443/thecircle2.jpg is tangent to the y-axis at point A(0,3) [see graph]. "a" is a parameter. Find the value of a. The Attempt at a...
  48. F

    Sample circle area following a distribution

    Hello! Hope someone can help me, I already tried some solutions, but couldn't come to a final conclusion. I need to create a circular area, from which particles are emitted. For this, I have a frequency distribution of x and y. I have around 100 points, but to simplify matters I list only...
  49. S

    Equations of a line tangent to a circle

    Homework Statement Given the circle (x+1)^2 + (y-3)^2 = 25, determine the equations of the tangents to the circle with the slope -3/4.Homework Equations y = mx + bThe Attempt at a Solution I thought that if I could find the equation of the line that passed through the center of the circle and...
  50. T

    Calculating Hydrostatic Force of a Submerged Semicircle

    Homework Statement There is a semicircle submerged in the water. The distance between the surface and the top of the semicircle is 2 ft, and the radius is 5 ft. Find the hydrostatic force. The answer is 1.2 * 10^4 N. Homework Equations y = sqrt(25 - x2) The Attempt at a Solution...
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