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. D

    Circle Inscribed in a Parabola?

    Homework Statement Find the largest circle centered on the positive y-axis which touches the origin and which is above y=x^2 Homework Equations equation of a circle: r^2=(x-a)^2+(y-b)^2 equation of a circle centered on the y-axis: x^2+(y-b)^2=r^2 equation of a parabola: y=x^2 The...
  2. E

    Is it possible to create a metal ring with a variable diameter using segments?

    I am interested to see if there is a way of creating a metal ring which has a variable diameter, the only way I imagine that this can be done is using segments which come together to form a circle, once pulled away the diameter changes however, the problem I face is that when the segments are...
  3. S

    Tangent to a Circle Homework Solution

    Homework Statement Given that the line ##y=mx+c## is a tangent to the circle ##(x-a)^{2} +(y-b)^{2} =r^{2}##, show that ##(1+m^{2}) r^{2}=(c-b+ma)^{2}##Homework Equations Quadratic discriminant, sum and product of rootsThe Attempt at a Solution I substituted y=mx+c into the equation of the...
  4. M

    Parametric equations for circle of curvature at given point.

    Hey guys, I'm new here. I got a problem from my professor that is different from any other problems we have done. I'm stuck and need a little help.Homework Statement r(t) = <cos(t), t, 2sin(t)> Find parametric equations for the circle of curvature at (0, pi/2, 2)The Attempt at a Solution I...
  5. P

    Solving Circle Equation with 2 Intersecting Vectors

    Thank you for taking time to read my post, I hope I am putting into the correct area of the physics forum. I am working with a programmer to complete a project that involves 2 intersecting vectors and a circle. The vector coordinates are known, we are trying to solve the circle equation...
  6. G

    Rate of change of area of circle in respect to radius

    What is the Rate of change of area of circle in respect to radius when radius is 3in I know that that dA/dr is equal to the circumference of the circle But where does that come from? Also the formula for the circumference of the circle is 2(pi)r But the answer is 6 (pi)in^2/in. I understand...
  7. S

    Equation of oscillating circle

    http://gyazo.com/bc7f6da4d4c4aca300bb5efb2410fc8d.png The problem, and my solution are in the image. I need help on finding the equation of the osculating circle, I found radius but I don't know where to go from there. Also if you could just check if my math seems correct that would help me...
  8. Saitama

    MHB Selecting three points on a circle

    (I am not sure if this is the right place to post it but since my solution involves some Calculus, I decided to post it here.) (Also, I am not well versed with the correct words and terms to be used while doing geometrical probability problems, I am sorry if I write something wrong. :o )...
  9. P

    Points on a Circle: Find Value of 'a

    Homework Statement For what values of a do the points (4,3),(-3,1),(1,a), and (1,5) lie on a circle? The Attempt at a Solution So my solution came down to getting the equation a(x^2+y^2)+bx+cy+d=0 Making a matrix with the first three points, doing R1-R4, R2-R4, R3-R4, then reducing the matrix...
  10. R

    MHB Calculating the Radius of a Graduated Circle

    Hello, I was working problems from a very old trigonometry book, Loney's Trigonometry from 1895. There appears here a problem stating: The value of the divisions on the outer rim of a graduated circle is 5' and the distance between successive graduations is .1 inch. Find the radius of the...
  11. B

    Why does Sin represent Y on unit circle

    As the title inquires, I am curious as to how or why the Sin function represents y coordinate on the unit circle.
  12. L

    MHB Maximizing the volume of a cone formed by cutting a sector from a circle

    From A circular sheet of RADIUS "R" a sector tie is cuts so that the coil Gets a funnel. Calculate the angle of the circular sector to cut back so of funnel has the maximum capacity. Answer tha angle is 2sqrt(6)pi/3
  13. T

    Electric Flux Through the Surface of a Circle

    Homework Statement A sphere of radius ## R ## carries charge density ## \rho = ar^5 ## where ## a ## is a constant. Find the flux ## \Phi ## of its electric field through a surface of a circle with radius ## R ## if the circle lies in a plane tangent to the sphere and its center touches the...
  14. K

    Finding base circle with coordinates of two points of involute curve

    Hi, I have an involute gear and measured co-ordinates of two arbitrarily chosen points (on the involute portion) of a tooth. Can I find out the base circle with this information? Thanks.
  15. N

    Computing tangential derivative d2x/ds2 at a point on a circle.

    Let P(x,y) be a point on a unit circle that is centered at (0,0). How to compute exactly the function \frac{\partial^2 x}{\partial s^2} where x is the x-coordinate of the point P(x,y) and s is the tangent at point P(x,y) . Clearly, \frac{\partial x}{\partial s} =...
  16. C

    Definition of a circle in point set topology.

    The circle seems to be of great importance in topology where it forms the basis for many other surfaces (the cylinder ##\mathbb{R}\times S^1##, torus ##S^1 \times S^1## etc.). But how does one define the circle in point set topology? Is it any set homeomorphic to the set ##\left\{(x,y) \in...
  17. H

    Energy of a system-a box moving in a circle

    Homework Statement A light spring has unstressed length 15.5cm.It is described by Hooke's law with spring constant 4.30N/m.One end of the horizontal spring is held on a fixed vertical axle,and the other end is attached to a punk of mass that can move without friction over a horizontal...
  18. Albert1

    MHB Area of Circle O: Calculate Here

  19. MarkFL

    MHB Robin's question at Yahoo Answers regarding the osculating circle of a parabola

    Here is the question: I have posted a link there to this thread so the Op can view my work.
  20. bananabandana

    Is a Sliced Circle Segment Equivalent to Half an Ellipse?

    Homework Statement Is it true that the if you slice a circle into two segments, you can think of one of the pieces as being half an ellipse? Homework Equations The Attempt at a Solution Not sure, I was just thinking about this! Not really any idea how to proceed to a solution, or how I can...
  21. MarkFL

    MHB Find Area of Lemniscate Bounded by Circle: r^2=6sin(2theta) & r=sqrt(3)

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  22. B

    Stress transformation, shear stress state, Mohr's circle c/work

    Homework Statement DIAGRAM ATTACHED AT BOTTOM Q. The following statements are true for an element in plane stress state. (this is 2D) (1) one of the principle stresses is 40Mpa; (2) σx= -2τxy; (the algebraic values) (3) in x'oy' with θ=30°, the two normal stresses σx'=σy' Determine...
  23. S

    Differentiating the Area of a Circle

    In my high school Calculus course, I've encountered several optimization problems involving the area of a circle and I noticed the obvious fact that if you differentiate the area of a circle you obtain the expression for its circumference. This implies that the rate of change of a circle's area...
  24. MarkFL

    MHB Find Tangent Lines Through Origin to a Circle

    Here is the question: I have posted a link there to this thread so the OP can see my work.
  25. K

    Circle Expansion on Expanding Sphere

    Homework Statement How would a circle on a sphere expand as a function of the sphere's radius as the sphere expands? Homework Equations none were provided The Attempt at a Solution \S=4\,\pi \,{R}^{2} A=\pi \,{r}^{2} {\it dA}=2\,\pi \,r{\it dr} {\it dS}=8\,\pi \,R{\it dR}...
  26. adjacent

    Proving the Chord One Rule in Circle Theorems | Isosceles Triangle Proof

    One rule of circle theorems is,a line drawn from center to the mid-point of a chord cuts the cord at 90°. What's the proof? It's true that two radius and a chord creates an isosceles triangle. So, How can I prove that in an isosceles triangle,a line drawn from the vertex angle to the mid-point...
  27. C

    Equation of the Osculating circle

    Homework Statement The ellipse has equations x = 2cos(t) and y = 3sin(t) where 0 <= t <= 2*pi The problem asked me to calculate the curvature at points (2,0) and (0,3). I did that, but now the problem asks what the equation of the osculating circle is at each of those points. I know the...
  28. Albert1

    MHB Find the radius of the small circle O_2

    find the radius of the small circle O_2:
  29. P

    Finding the Center of Mass of a Semicircle

    Ok so when I try to get the centroid of a semicircle using ∫rdm/∫dm, ∫dm = ρ∫dxdy for x^2, but now that the area is (0.5)∏(x^2 + y^2) what would ∫dm be? If I did use ρ∫dxdy I have to do some weird integration of sqrt(30^2 - x^2) which we haven't learned yet. If I do have to ρ∫dxdy how would I go...
  30. Saitama

    Selecting things arranged around a circle

    Homework Statement If n things are arranged in circular order, then show that the number of ways of selecting four of the things no two of which are consecutive is $$\frac{n(n-5)(n-6)(n-7)}{4!}$$Homework Equations The Attempt at a Solution My approach to the problem is that I first select the...
  31. D

    Finding Tangent Equation from (-5,4) to Circle

    Homework Statement Find the equation of the tangent drawn from the point(-5,4) to the circle x^2+y^2-2x-4y+1=0 Homework Equations y-y1=m(x-x1) x^2+y^2+2gx+2fy+c=0 The Attempt at a Solution The equation of the tangent using the point is as follows y-4=m(x+5). Now, if I substitute...
  32. G

    Bivariate density on a unit circle

    Homework Statement Consider the bivariate density of X and Y, f(x, y) = pi/2 for x^2 + y^2 ≤1 and y > x and = 0 otherwise. (a) Verify that this is a bivariate density (that is, the total volume ∫∫ f(x,y)dxdy = 1) Homework Equations The Attempt at a Solution The problem I'm having is...
  33. Albert1

    MHB Find Radius of Circle: Calculate Here

  34. D

    Finding the Length of a Chord on a Circle

    Homework Statement Find the length of the chord which the circle 3x^2+3y^2-29x-19y+56=0 cuts off from the straight line x-y+2+=0. Find the equation of the circle with this chord as diameter Homework Equations x^2+y^2+2gx+2fy+c=0 The Attempt at a Solution I can solve the second part...
  35. D

    How to Find the Equation of a Circle with Given Points and Radius?

    Homework Statement Find the equation of the circle which intersects the x-axis at two points at 2 unit distance from the origin and the radius is 5 Homework Equations x^2 + y^2 + 2gx + 2fy + c=0 r=(g^2+f^2-c)^1/2 The Attempt at a Solution I first tried to solve it by assuming...
  36. K

    What Size Circle Fits Between Three Touching Circles?

    Hi -- Thinking about the problem, where I have three circles in a closest possible packing inside an equilateral triangle. So two circles on the floor, adjacent to each other touching and a third circle placed on top so that the distances off their centers from each other are all 2R, R=Radius...
  37. P

    MHB Can three tangents to a circle meet at a common point?

    I considered the question of whether three tangents to the circle could meet at a common point and only came up with a contradiction by lengthy constructive means. Circles are "nice", so there must be some clever ways of showing this fact that given a point outside the circle there are two...
  38. I

    Use vectors to form a right triangle on a circle

    Homework Statement Use vectors to demonstrate that on a circle any two diametrically opposed points along with an arbitrary third point(on the circle) form a right triangle Homework Equations Hint: assume without a loss of generality that the circle is centered at the origin and let v...
  39. J

    How Many Unique 5 White and 5 Black Bead Necklaces Can Be Made?

    Homework Statement How many necklace with 5 white beads and 5 black beads can be constructed? Homework Equations Circular Permutation problem The Attempt at a Solution] I did 10!/5!5!=252 but from there I didn't get anywhere. I know this includes repeats from rotational...
  40. T

    Conservation of energy in a circle arc.

    http://www.natuurkunde.nl/servlet/supportBinaryFiles?referenceId=1&supportId=606217 Hello everyone, I was wondering if anyone could shed some light on the following problem: While composing a practise test for a chapter about conservation of energy, I made a problem like the one in the...
  41. O

    A block spins around a circle with thrust

    Homework Statement A 400g steel block rotates on a steel table while attached to a 1.20m -long hollow tube. Compressed air fed through the tube and ejected from a nozzle on the back of the block exerts a thrust force of 4.71N perpendicular to the tube. The maximum tension the tube can...
  42. MarkFL

    MHB Find Area Between Circle & Function: Calc II

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  43. S

    MHB Equation of a Circle in the Complex Plane

    Hi Guys can you please help me out for the following question: Show that the equation of the circle $$\gamma(a;r)$$ centered at $$a\in\mathbb{C}$$ and radius $$r$$ can be written in the form: $$|z|^2 - 2Re(\bar{a}z) + |a|^2 = r^2 $$
  44. E

    Why the curve r(t) approaches a circle as t approaches infinity

    Both statements 1 and 2 are given as an explanation of why the original statement is true, but I don't understand why you can use statement 2 (since in the original vector equation you do not have Sin2(t), -Cos2(t)) Show why r(t) = <e-t, Sin(t), -Cos(t)> approaches a circle as t →∞. 1. As...
  45. G

    Question about turning a hemisphere into an equivalent circle?

    I was thinking about this when I was trying to work out a simpler way of finding the volume of a sphere. Suppose we cover a hemisphere with a piece of pliable thin cover. Stretching the cover flat would make a circle. The circumference of the sphere is 2*pi*r. The distance along the surface of...
  46. jegues

    Power Circle Diagram: Sending/Recieving End Voltages & Power Angle

    Homework Statement How does the angle and magnitude difference between the two ends of the transmission line effect the real and reactive power flow? Homework Equations The Attempt at a Solution See two tables of data points and plots attached. It seems as though for the...
  47. M

    A particle is moving in a circle

    A particle is moving in a circle of radius R in the xy plane. During the motion, neither the x- nor y-component of the particle's velocity exceeds v. Find the minimum possible time for the particle to complete one circle. (I.e., find the minimum possible period of revolution.) Hint 1: I...
  48. L

    Curve of a Circle: Find the Equation

    Homework Statement Find the curve whose curvature is 2, passes through the point (1,0) and whose tangent vector at (1,0) is [1/2 , (√3)/2 ]. The Attempt at a Solution I know I must use the Fundamental theorem of plane curves but I don't know how to apply it correctly here. Another...
  49. PsychonautQQ

    Finding center of circle with Polar Coordinates

    Homework Statement r=7sin(∅) find the center of the circle in Cartesian coordinates and the radius of the circle The Attempt at a Solution My math teacher is impossible to understand >.< and then the stupid homework is online and crap blah this class but I REALLY want to understand the material...
  50. E

    What is the approximate uncertainty in the area of a circle

    Homework Statement Approximate uncertainty in the area of a circle of radius 5.5×10^4 cm? Homework Equations πr^2 The Attempt at a Solution
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