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. Monoxdifly

    MHB Circle Equation: Solving for x^2 + y^2 + Ax + By = 0 w/y=4/3x

    A circle L is going through the point O (0, 0) and P (6, 0). The center is in the line y=\frac{4}{3}x. The equation of the circle L is ... A. x^2+y^2+6x-8y=0 B. x^2+y^2-6x-8y=0 C. x^2+y^2-8x-6y=0 D. x^2+y^2+8x+6y=0 E. x^2+y^2-4x-3y=0 Since the equation of a circle is x^2+y^2+Ax+By+C=0, I...
  2. Physiona

    Quadratic & Equation of a Circle

    Homework Statement Two equations are defined as follows: What is the value of ?Homework Equations Quadratic Format: ##ax^2+bx+c=0## ##y^-x## = ##1/y^x## The Attempt at a Solution I'm not sure In how to attempt this style of question as I know quadratic and equation of...
  3. J

    Interpreting Dip Circle Measurements: True or Apparent Dip?

    Homework Statement Homework EquationsThe Attempt at a Solution I know how to solve this problem . I just need help in interpreting the language of the problem as to what is given and what is required to calculate . 1) Is 30° the true dip and we are asked to calculate apparent dip OR 2) Is...
  4. Z

    Fluid Dynamics -- Use the Milne-Thomson circle theorem to show the complex potential for a fluid....

    Homework Statement Two equal line sources of strength k are located at x = 3a and x = −3a, near a circular cylinder of radius a with axis normal to the x, y plane and passing through the origin. The fluid is incompressible and the flow is irrotational (and inviscid). Use the Milne-Thomson...
  5. D

    Energy analysis of a particle moving in a shrinking circle

    Homework Statement A particle of mass m is moving on a frictionless horizontal table and is attached to a massless string, whose other end passes through a hole in the table, where I am holding it. Initially the particle is moving in a circle of radius ##r_0## with angular velocity ##w_0##, but...
  6. Physiona

    Circle Geometry with an Intersecting Line

    I'm requiring help on a circle geometry question I've done. The line L, has equation of y=0, and intersects the circle with (3,0) and radius of 29. Find the points of intersection. My working out: 292 = 841 It's centre is 3,0, Inserting that in circle equation gives (x-3)2+y2 = 841 Solving...
  7. Ventrella

    A Binary fractal tree with equidistant leaves on a circle

    Does there exist a binary fractal tree… (reference: http://ecademy.agnesscott.edu/~lriddle/ifs/pythagorean/symbinarytree.htm ) …whose leaves (endpoints) lie on a circle and are equidistant? Consider a binary fractal tree with branches decreasing in length by a scaling factor r (0 < r < 1) for...
  8. claraberner

    Mohr's Circle does y-axis = max strain or max strain/2?

    This isn't a problem assignment per-say, but for a lab calculation. In my mechanics of material's class, we learned that the radius of Mohr's circle is the maximum strain, but now in my application lab-based class, the video is saying the radius is equal to the maximum strain divided by 2. So...
  9. Observeraren

    I Turning the square into a circle

    Hello Forum, Does topology reckon the art of turning a square into a circle? I am quite new to topology and maths in general, I have only dabbled and eyed on my collection of mathbooks. I have come to a conclusion of how to turn the Square into A Circle without cutting. I wonder if I am...
  10. Ghost Writer

    Exploring Turning Circle Differences in Ancient Chariots

    Hi all - In the Near Eastern Late Bronze Age the Egyptians employed a two-man chariot with the axle placed at the rear of the vehicle; whereas the Hittites employed a three-man chariot with the axle placed in the center of the vehicle. As a result, the weight of the two-man crew in the Egyptian...
  11. alijan kk

    Equation of Circle Passing Through Given Point and Line with Given Radius

    Homework Statement Find an equation of the circle passing through: A(-3,1) with radius 2 and centre on the line 2x-3y+3=0 Homework Equations x2+y2+2gx+2fy+c=0 r2=g2+f2-c The Attempt at a Solution using this equation , i have found 2 equations -6g+2f+c=-10 by putting (-3,1) -2g+3f+3=0...
  12. F

    B Forces acting when turning a circle on a unicycle

    Hello. Lets say I am riding a unicycle rolling along the wheels direction with some velocity v0. I know that for the unicycle to turn a curve the centripedal force must be provided by the static friction with the rode that acts against any slipping of the wheel. I am however having trouble...
  13. Y

    MHB Equilateral triangle within a circumscribed circle

    Dear all, In the attached picture there is an equilateral triangle within a circumscribed circle. MW is a radius of the circle, and I wish to prove that MT = TW, i.e., that the triangle cuts the radius into equal parts. I thought perhaps to draw lines AM and AW and to try and prove that I get...
  14. talknerdy2me

    A ball moves a horizontal circle....

    Homework Statement While a string is attached to a ball and the ball moves in a horizontal circle at a constant speed, does the force change both the direction and speed of the ball? Explain. Homework Equations No equations necessary. The Attempt at a Solution *I think* the force changes...
  15. J

    What are the dimensions and boundaries for different types of integrals?

    Homework Statement [/B]Homework Equations Substitution. The Attempt at a Solution Since the circle is of unit radius and around origin, limits are x = -1 to 1, and y = -1 to 1 I replaced x by cos t, and y by sin t. But what to put in place of ds? I thought about divergence theorem, but then...
  16. K

    Triangle inscribed in a circle

    Homework Statement [/B] In a circle with center S, DB is the diameter. The line AC goes 90 degrees from the center point M of the line SB. " What type of triangle is ACD? 2. Homework Equations The Attempt at a Solution I can see it is an equilateral triangle, but do not know how to explain...
  17. S

    MHB AO+BO+CO≥6r where r is the radius of the inscribed circle

    From the entrance examinations to Ghana University ,from high school, i got the following problem: If O is the center of the inscribed circle in an ABC trigon,then prove that: AO+BO+CO\geq 6r where r is the radius of the inscribed circle.
  18. K

    Centripetal motion, the tension at the bottom of a circle

    Homework Statement A mass m, at one end of a string of length L, rotates in a vertical circle just fast enough to prevent the string from going slack at the top of the circle. Assuming mechanical energy is conserved, the tension in the string at the bottom of the circle is: a) 6 mg b) mg +...
  19. L

    Acceleration of a speck around a circle

    Homework Statement a grinding wheel 0.5 m in diameter roates at a rate of 8.00 x 10^2 revolutions per minute. find the magnitude of the acceleration of a speck of metal cuaght in the outer edge of the wheel Homework Equations a=4pi^2r/T^2 [/B]The Attempt at a Solution I was wondering if i...
  20. K

    Circular motion, string and ball in a horizontal circle

    A mass m = 0.15 kg is attached to a massless string and rotates at constant speed v = 4 m/s in a horizontal circle of radius 2 m. The tension T (in N) in the string is: (a) 1.1 (b) 1.9 (c) 2.4 (d) 3.3 (e) 4.9 I would assume that first I calculate the centripetal acceleration by using v^2/r =...
  21. Pushoam

    Flux of a point charge through a circle

    Homework Statement Homework EquationsThe Attempt at a Solution I will try to choose the correct option using the common sense instead of solving it. As d decreases, the flux should increase. For R>>d, only option (a) and (d) satisfy this condition. Now, for choosing between (a) and (d)...
  22. Spinnor

    Riding a motorcycle in a circle, power slide

    Say you (a skilled motorcycle racer) ride a motorcycle in a big empty parking lot and ride in a big circle of constant radius and slowly go faster and faster. If you are careful as you go faster will you all of a sudden go from very little power slide to a lot of power slide like the video...
  23. M

    Centripetal force throughout a vertical circle

    Homework Statement In what position in vertical circular motion is the centripetal force the greatest? Top, Bottom, Left, or Right Homework Equations Can someone explain how Fc is greatest at the top? The Attempt at a Solution I had reasoned that since centripetal acceleration which I will...
  24. J

    What is the acceleration normal to PC in polar coordinates?

    Homework Statement Homework EquationsThe Attempt at a Solution Since the force is always directed towards C , angular momentum about C should be conserved . But that doesn't seem to help as we need the relation at any general angle . How should I proceed ?
  25. C

    B Calculating the area of a circle or square using decimals

    I came across something that is completely counter-intuitive, and I'm wondering if I'm correct or not. If a square has a side that is .8m someone would do .8 time .8 which is .64. How can an area be smaller than a side I thought and so I looked it up and found only one site that said something...
  26. M

    MHB How to Determine the Equation of a Circle from Given Diameter Coordinates?

    Determine the equation of the circle in standard form, given the coordinate of the diameter PQ. P(-4, -2) and Q(6, 4) Midpoint is (5, 3). d = sqrt{(6-(-4))^2 + (4-(-2)) d = sqrt{(10)^2 + (6)^2} d = sqrt{100 + 36} d = r = sqrt{136} Let d = distance = radius (x - h)^2 + (y - k)^2 = r^2...
  27. O

    Lagrangian of system with circle and cube

    Hello. I have some problems with making Lagrangian. I need your advice. 1. Homework Statement I have this situation: Consider the circular path is intangible and without friction. I have to find Lagrangian for coordinates x and θ. Homework Equations [/B] L = U - V The Attempt at a...
  28. M

    MHB Finding the Equation of a Circle Given the Diameter Coordinates

    Determine the equation of the circle in standard form, given the coordinate of the diameter PQ. P(1, -3) and Q(-5, -5)
  29. M

    MHB What caused the wrong answer for determining the center and radius of Circle 2?

    Determine the center and radius of circle.
  30. M

    MHB What is the Center and Radius of a Circle?

    A. Determine the center and radius of circle. B. Also, find the y-coordinates of the points (if any) where the circle intersects the y-axis.
  31. A

    Angular Velocity in Simple Harmonic Motion

    I am very confused about angular velocity ω and why its used in simple harmonic motion. ω is described as θ/τ but when it comes to masses on springs, there is no angle - it is zero. Angular velocity comes from circular motion but the motion of SHM is not circular. My confusion is even greater...
  32. S

    I Average chord length of a circle

    I would like to find the average chord length of a circle. And I have 2 methods, which gave different answers... [The chord is defined as the line joining 2 points on the circumference of the circle.] The general formula for a chord length is ##d=2R\sin(\delta/2)=2\sqrt{R^2-u^2}## Method 1...
  33. S

    Mapping Circle to Ellipse with Dilation?

    Homework Statement The problem comes from S. Lang's "Basic mathematics", chapter 7, §1: "Consider the following generalization of a dilation. Let ##a > 0, b > 0##. To each point ##(x, y)## of the plane, associate the point ##(ax, by)##. Thus we stretch the x-coordinate by ##a## and the...
  34. A

    MHB Mapping of a Circle in the Complex Plane

    I have a circle with centre (-4,0) and radius 1. I need to draw the image of this object under the following mappings: a) w=e^(ipi)z b) w = 2z c) w = 2e^(ipi)z d) w = z + 2 + 2i I have managed to complete the question for a square and a rectangle as the points are easy to map as they are...
  35. F

    Finding the period of an object moving in horizontal circle

    Homework Statement A small mass m is suspended from a string of length L. The body revolves in a horizontal circle of radius R with a constant speed v. Find the speed of the body and the period of the revolution. Homework Equations ΣFx = Tx = mV^2/r Period = (2πr)/V The Attempt at a...
  36. A

    Finding the direction of an angle in the unit circle

    Homework Statement I'm having trouble understanding how to find the angle of a vector. Here we are given the x and y component to help us find the direction of vector C. In this case, both x and y component is negative, so it should be in the third quadrant. I know that since we have both the x...
  37. starstruck_

    Calculating Acceleration of a Toy Plane Flying in a Circle

    Question: Starting from rest, the toy plane flies around a circle of radius 2m, three times in 3 seconds. There is constant tangential acceleration, fins the magnitude of the acceleration at the end of 0.5s. My solution (most likely wrong): Circumference= 2pir= 4pi= 12.56631 Distance...
  38. PsychonautQQ

    I Impossible to lift the identity map on the circle

    Suppose that L: ##S^1## ---> ##R## is a lift of the identity map of ##S^1##, where e is the covering map from ##R## to ##S^1##, where ##R## is the real numbers and ##S^1## is the circle. Then the equation e * L = ##Id_{S^1}## (where * is composition) means that 2*pi*L is a continuous choice of...
  39. lfdahl

    MHB What is the Sum of Lengths for a Regular n-gon Inscribed in a Unit Circle?

    Let $S_n$ be the sum of lengths of all the sides and all the diagonals of a regular $n$-gon inscribed in a unit circle. (a). Find $S_n$. (b). Find $$\lim_{{n}\to{\infty}}\frac{S_n}{n^2}$$
  40. R

    How does the kinetic energy of a swinging ball change with respect to height?

    Homework Statement Imagine a ball on a string that we swing vertically so that the hight changes. By conservation of energy the velocity of the ball must change right? Because at the highest point of the swing it will have maximum GPE but at the bottom, minimum right? Watching many videos has...
  41. Ventrella

    A Find a circle inside of and tangent to a larger circle

    Start with a circle of radius r and center c. Inside of that circle is an arbitrary point p. Given an arbitrary normalized direction vector d, I need to find the radius and center the circle that (1) intersects p, (2) is tangent with the circle centered at c, and (3) has its center lying on the...
  42. G

    General solution for the heat equation of a 1-D circle

    Homework Statement Modify the initial conditions (for the diffusion equation of a circle) to have the initial conditions ## g(\theta)= \sum_{n=-\infty}^{\infty}d_{n}e^{2\pi in\theta} ## Using the method of Green's functions, and ## S(\theta,t)= \frac{1}{\sqrt{4\pi...
  43. Mr Davis 97

    I Why is this line not homeomorphic to the unit circle?

    I've been told that ##[0, 2 \pi )## is not homeomorphic to the unit circle in ##\mathbb{R}^2##. Why not? From intuition, it would seem that I could just bend the line segment to fit the shape of a circle.
  44. karush

    MHB 15.3.50 Double integral of circle and graph

    $\displaystyle \int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \sqrt{x^2+y^2} \, dydx=\frac{\pi}{6}$ this was the W|A answer but how ? also supposed to graph this but didn't know the input for desmos
  45. D

    Coefficient of friction between stick and circle

    Homework Statement Using the results from problem 2.18 for the setup shown in the Figure below show that if the system is to remain at rest, then the coefficient of friction:a) between the stick and the ground must satisfy $$ μ ≥ \frac {sin(Θ)cos(Θ)} {(1+cos(Θ))(2-cos(Θ))} $$ Homework...
  46. karush

    MHB 243.11.5.9 Area of intersection cardioid and circle

    OK just seeing if this is setup OK before I pursue all the steps I thot adding areas would be easier:cool:
  47. A

    Finding the approximate change in the perimeter of a circle

    Homework Statement The radius of a circle increases from 3 to 3.01 cm. Find the approximate change in its perimeter. Here's a link to the actual question, in case you need the answer for 6(a) to solve 6(b) http://imgur.com/a/nQt6M Homework Equations Perimeter of circle = 2πr Area of circle =...
  48. M

    MHB Equation of Circle in Standard Form

    Determine the equation of the circle that passes through the point (-4, 1) and its center is the midpoint of the line segment joining the centers of the two circles given below: x^2 + y^2 -6x - 4y + 12 = 0 and x^2 + y^2 - 14x + 47 = 0. Write the equation in standard form. 1. Does the...
  49. M

    MHB Equation of Circle Through Origin

    Find the equation of the circle passing through the origin and centered at the point (3,5). Origin means the point (0,0). From the previous example, I found the equation of the circle centered at (3,5) to be (x - 3)^2 + (y - 5)^2 = 25. I do not understand what part the origin plays here. The...
  50. M

    MHB Finding the Equation of a Tangent Circle with Center (3,5)

    Find the equation of the circle tangent to the x-axis and with center (3,5). Can someone get me started? I know this circle touches the line y = 0 and its center point (3,5) lies in quadrant 1.
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