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. Pushoam

    Maximum distance the car accelerates in a circle without skidding

    Homework Statement Homework EquationsThe Attempt at a Solution At distance s, the speed of the car is v. $$ v^2 = 2wτs$$ $$\frac { mv^2} R ≤ kmg$$ Let's denote the maximum distance covered without sliding is smax. $$\frac { m2wτsmax} R = kmg$$ $$ smax = \frac {kgR} {2wτ}$$ Is this correct...
  2. MrsTesla

    B Sidereal Time and the Arctic Circle

    This may be a basic question, but why does the Sun rise at the same sideral time between December 22 and June 22 on the Arctic Circle? And how can I prove it?
  3. M

    What is Implicit Differentiation for a Circle?

    Homework Statement Hello I have this circle with the equation : [/B] (x-a)^2+(y-b)^2=r^2 I want to find dy/dx for it 2. Homework Equations (x-a)^2+(y-b)^2=r^2 The Attempt at a Solution I am looking on the internet and it appears that I should use what is called "Implicit differentiation"...
  4. H

    How to find chords, intersections of chords on circle?

    Homework Statement Homework EquationsThe Attempt at a Solution this is the answer https://www.algebra.com/algebra/homework/Circles/Circles.faq.question.1038060.html but why the c1 = 0, c2 = 1, c3 = 3, c4 = 6 etc why not c2 = 2? c4 = 4?
  5. victorhugo

    Can you find orbital velocity from circle equation y^2+x^2=r^2?

    Maintaining R as the constant hypotenuse in the triangle formed by x and y coordinates in a 'perfect' circle, r2=x2+y2 r2=x2+y2 So knowing that in 9.8 metres above ground it will take 1 second for an object to fall, I tried to find how many metres in the X direction an object must cover in 1...
  6. Pouyan

    Potential of a circle boat

    Homework Statement A line charge has the total charge Q evenly distributed over a circle boat with radius a and sector 2β, placed according to the figure Find the Electric field E and the potential V in the origin. Homework Equations I know for this case that E(r) = (1/4πε) ∫ (λ(r')/R2)R...
  7. H

    How to find the angles of a triangle in a semicircle?

    Homework Statement Homework Equations d(y)/d(x) --> max area area of triangle = 1/2 . base . height The Attempt at a Solution for number (2) [/B] x^2 + y^2 = r^2 --> circle equation base = 2R, height = y Area = 1/2 . 2R . y area = 1/2 . 4. √ (r^2 - x^2) area now is half of max = 2, so...
  8. H

    A circle is circumscribed around triangle ABC, find length?

    some formula related I tried to draw the problem can anyone give me clue how to solve it?
  9. davidge

    I Is it possible to prove that the circle is a manifold using open spheres?

    The answer to the question of the thread title is yes, according to what I found on web. Now a manifold is by definition a topological space that (aside from other conditions) is locally Euclidean. What does such condition means? Is it the same as saying that each of its points must have a...
  10. H

    Circle Problem: Solve an angle

    Homework Statement [/B] imgur.com/a/bsAKl [I didn't know how else to upload an image from my iPad] In this problem: -Center of the circle=Point F -Arc AB= 110 degrees -Arc CD= 40 degrees Find the measure of angle E. Homework Equations I know of some equations, but I don't know if they...
  11. M

    MHB Finding the Equation of a Circle Given Specific Conditions

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

    MHB Equation of Circle Through Origin

    Find the equation of the circle passing through the origin with center (3, 5). Can someone get me started? Must I use the point (0, 0) here?
  13. M

    MHB What is the equation of a circle tangent to the x-axis with a center at (3, 5)?

    Find the equation of the circle tangent to the x-axis and with center (3, 5). (x - h)^2 + (y - k)^2 = r^2 h = 3, k = 5 r = 5 (x - 3)^2 + (y - 5)^2 = 5^2 (x - 3)^2 + (y - 5)^2 = 25 Yes?
  14. davidge

    I Is the Unit Closed Disk Minus the Origin Homeomorphic to the Unit Circle?

    The unit closed disk minus the point ##(0,0)## ##\mathbb{D}^1 \setminus (0,0): \bigg[(x,y) \in \mathbb{R}^2 | 0 < x^2 + y^2 \leq 1 \bigg]## is homeomorphic to the unit circle ##\mathbb{S}^1: \bigg[(x,y) \in \mathbb{R}^2 | x^2 + y^2 = 1 \bigg]## Since ##\mathbb{D}^1 = \big(\mathbb{D}^1 \setminus...
  15. M

    MHB How Do You Determine If a Point Lies on a Circle?

    1. Sketch the circle of radius 1 centered at (0, 0). (A) Write the equation of this circle. I must use x^2 + y^2 = r^2. The radius is 1. This means r = 1. The equation is x^2 + y^2 = 1. Correct? B. Does the point (3/5, 4/5) lie on this circle? (3/5)^2 + (4/5)^2 = 1^2 (9/25) + (16/25) = 1...
  16. P

    MHB Tilting a circle or ring over backwards creates an ellipse....

    What I would like to be able to calculate is the following: Suppose a hoop or ring is held up perpendicular to the ground and you stood in front of it, it would look perfectly circular and knowing the radius you could calculate the area of this circle. Now if this hoop was tilted over backwards...
  17. B

    MHB Combinatorics problem : circle hopscotch

    Can anyone help me with the following scenario: A hopping circuit is painted on a school playground. It consists of 25 small circles, with the numbers 0 ( at the 12 o' clock place) to 24, arranged as a big circle. Each student jumps either 3 or 4 spaces clockwise(so a student can end up either...
  18. Monoxdifly

    MHB Segments in a Circle: How Many Parts Can Form & What Will It Look Like?

    If a circle has 6 segments, how many maximum parts which can be formed? I know that 1 segment makes 2 parts, 2 segments make 4 parts, and 3 segments makes 7 parts. Judging by the pattern, is the answer 22? What will the exact picture of the circle be? Thank you very much.
  19. D

    B Explain FBD in vertical circle

    Hi , just curious about the F.B.D of a pebble moving in a vertical circle, which component that balances the weight of the pebble at the horizontal position , as tension is providing the required centripetal force , weight is acting downwards , which component balances it ?
  20. jamalkoiyess

    MATLAB How to create distinct circles in Matlab?

    I want to create a plate of distinct circles on Matlab where their radii are generated by randn(1,p) and centers are random. I am currently doing the circles using viscircles, but some of them are overlapping, and since I want approximately 100 ones, this problem only gets worse. How can I make...
  21. D

    MHB Can you find x using the trigonometry of circle sectors?

    I need help in solving this problem. Below shows all the measurements of the diagram, I need to find x:
  22. M

    B How Does the Unit Circle Relate to Euler's Formula in Complex Numbers?

    Hi everyone. I was looking at complex numbers, eulers formula and the unit circle in the complex plane. Unfortunately I can't figure out what the unit circle is used for. As far as I have understood: All complex numbers with an absolut value of 1 are lying on the circle. But what about...
  23. Mario

    Circles and Euler spiral (repost from general math)

    Hi, i have this problem..., giving two circle (example radius 1 = 500 units, radius 2 = 200 units, distance between centers = 275.73 units) find Euler Spiral (aka Cornu spiral, aka Clothoid https://en.wikipedia.org/wiki/Euler_spiral) tangent giving circle (unknown tangent points). For this...
  24. Feodalherren

    Frictionless ring sliding on circle in constant acceleration

    Homework Statement A box is sliding with constant acceleration a to the right. Inside the box there is a quarter of a circle upon which a frictionless ring can slide. Find the angle theta in terms of the other given variables. Picture in solution Homework Equations F=ma, etc. basic stuff...
  25. B

    Equilibrium of Charges at Circumference of a Circle

    Homework Statement Two charges placed at circumference of a circle of radius ##a## at ##\pi/2## from each other. Find the relative magnitude of third charge kept on the circumference such that the system is at equilibrium.Homework Equations Coulombs law. The Attempt at a Solution Let ##Q##...
  26. dextercioby

    B Can You Solve This Simple Geometry Problem with Just a Ruler and Pencil?

    Yes, one more reason to be humble, I know. This is the simplest problem I couldn't solve so far. Assume we have a circle of center O, a ruler of arbitrary size and a pencil. We use the ruler and the pencil to choose 4 points on the circle - the extremities of two diametral/diagonal segments...
  27. G

    B Transforming a circle onto another plane

    I am putting in a new flue for a Pot Belly stove. The ceiling through which the flue goes slopes at 12.5º. I want to cut a hole in a metal plate that will be fixed to the ceiling, and through which the flue passes. So, a vertical circular flue, radius R, passing through a ceiling sloped at...
  28. O

    Calculate circle radius with segment height and perimeter

    (mentor note: posted in a non-homework forum hence no template) Hello! I have a problem I'm trying to solve. I'm transforming a circle with known radius. Knowing it's radius i can calculate the circumference. I transform it by squeezing one side, leveling it, creating a circle segment with a...
  29. M

    MHB Find Center and Radius of Circle

    Determine the center and the radius of circle. x^2 + y^2 - 10x + 2y + 17 = 0 Can this be done using completing the square?
  30. R

    Tension in string when ball is at the top of a circle

    Homework Statement A child is swinging a .325 kg ball at the end of a .74 m long string in a vertical circle. string can withstand a tension of 12 N before breaking. What is the tension in the string when the ball is at the top of the circle if its speed at that point is 3.4 m/s? Homework...
  31. M

    MHB How can language proficiency affect understanding of math concepts?

    The radius of a circle is r units. By how many units should the radius be increased so that the area increases by b square units? I don't know where to begin. A = πr^2 Does this question involve the area of a circle formula? If so, in what way?
  32. R

    I Arc Length Parameterization for Unit Circle: Cos(s) & Sin(s)

    (cos(s), sin(s)) gives an arc-length parameterization of the unit circle so that the speed is constantly 1, but the second derivative doesn't give zero acceleration which should be the case with constant speed?
  33. L

    B Is There an Arctic Circle on the Moon?

    Do the lunar poles have something like and Artic circle where the Sun and/or Earth are fully visible in the sky for very long periods of time?
  34. Albert1

    MHB Incircle and circumscribed circle prove :d=√(R(R−2r))

    $\triangle ABC$ with its incircle $I$ (radius $r$) and circumscribed circle $O$ (radius $R$) the distance between points $O$(circumcenter) and $I$(incenter) is $d$ prove:$d=\sqrt {R(R-2r)}$
  35. M

    I Thin film around circle and ignoring curvature

    Hi PF! If we have flow around a curved object that is sufficiently thin, I Have seen many texts assume the surface is linear rather than curved. Can someone help me with what "sufficiently thin" is quantitatively and how this allows us to neglect surface curvature? As a simple toy problem...
  36. L

    What is the equation of the tangent line to a circle passing through the origin?

    Homework Statement Equation: x^2+y^2-6x-2y+8=0 Find the center and the radius. (Help) : Find the equation of the tangent to the circle above that passes through the beginning of axis O (0,0)The Attempt at a Solution I found the center and radius and i believe the values are : C (3,1) and R...
  37. Anatalbo

    Normal forces for small car performing a vertical loop

    Homework Statement A small car with mass .800 kg travels at a constant speed of 12m/s on the inside of a track that is a vertical circle with radius 5.0m. If the normal force exerted by the track on the car when it is at the top of the track is 6.00N, what is the normal force at the bottom of...
  38. Khunpol Jermsiri

    Circular motion object falling in the circle

    Homework Statement An object with mass of m traveling in a circle rail ,when reached point A the object derailed and moved in a parabola path to point B and so on find the length of AB in term of R and theta. given that B is on the same horizontal level as A Homework EquationsThe...
  39. M

    Eigenvalues and Eigenfunctions in Solving 2D Wave Equation in a Circle

    Homework Statement Solve 2D wave eq. ##u_tt=c^2 \nabla^2u## in a circle of radius ##r=a## subject to $$u(t=0)=0\\ u_t(t=0)=\beta(r,\theta)\\u_r(r=a)=0\\$$and then symmetry for ##u_\theta(\theta=\pi)=u_\theta(\theta=-\pi)## and ##u(\theta=\pi)u(\theta=-\pi)##. Homework Equations Lot's I'm sure...
  40. J

    Kinetic Energy in Circular Motion: Is 1/2 m v^2 Still Applicable?

    Homework Statement in circular motion (e.g. a pendulum) is the kinetic energy still 1/2 m v ^2 or is it a different equation? Homework Equations 1/2 m v ^2 The Attempt at a Solution
  41. M

    Show curvature of circle converges to curvature of curve @ 0

    Homework Statement Let γ : I → ℝ2 be a smooth regular planar curve and assume 0 ∈ I. Take t ≠ 0 in I such that also −t ∈ I and consider the unique circle C(t) (which could also be a line) containing the 3 points γ(0), γ(−t), γ(t). Show that the curvature of C(t) converges to the curvature κ(0)...
  42. S

    Calculating Force/Moment to Bend Strip into Circle

    I need to calculate the force or moment required to bend a thin strip into a circle
  43. M

    MHB Equation of the Circle (Part 2)

    Find the equation of the circle tangent to the y-axis and with center (3, 5). Can someone provide the steps needed to solve this problem?
  44. M

    MHB What is the equation of the circle tangent to the x-axis and with center (3, 5)?

    Find the equation of the circle tangent to the x-axis and with center (3, 5). Can someone provide the steps needed to solve this problem?
  45. W

    How Does Angular Speed Relate to Velocity in Concentric Circle Streamlines?

    Homework Statement The streamlines of a certain flow are concentric circles about the origin, and the absoute value of the velocity varies according to the law |V|=k*r^(n) Show that the angular speed of any fluid element in flow is described by: εz=(1/2)*k*(n+1)*r^(n-1) Homework Equations...
  46. R

    Circle inscribed in a triangle exercise

    Homework Statement In the drawing you can see a circumference inscribed in the triangle ABC (See the picture in the following link). Calculate the value of X https://goo.gl/photos/CAacV2dJbUrywfXv92. The attempt at a solution It seems I found a solution for this exercise with the help of a...
  47. T

    Confused about tangent to a circle (polar/Cartesian convert)

    Homework Statement The problem is from D'Inverno's book on GR, problem 5.6. We're using the Jacobian/transformation matrix to convert the tangent to a circle centered at the origin of radius A from Cartesian to polar coordinates. I can do the problem and get the book answer, that's okay...
  48. C

    Rubber Band Spinning in midair

    Homework Statement A uniform thin circular rubber band of mass M and spring constant k has an original radius R. Now it is tossed into the air. Assume it remains circular when stabilized in air and rotates at angular speed ω about its center uniformly. Derive an expression for the new radius...
  49. P

    General Relativity and the Circumference of a Circle

    Homework Statement Calculate the circumference of the circle θ = θ0 (a constant) in the spatial geometry \begin{eqnarray*} dS^2 = a^2(d\theta^2 + sin^2\theta cos^2\theta d\phi^2) \end{eqnarray*} Hence, (by finding R(z)) sketch the cross section of the surface embedded in three dimensions via...
  50. Biker

    Induced EMF in a circle: Faraday's law

    Homework Statement Explain what happens in the following situation: You have a loop in the form of a circle and there is a varying magnetic field inside that loop as the following picture illustrates: Homework Equations Faraday's law lorentez force The Attempt at a Solution I first thought...
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