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. N

    Standard Form of the Equation of a Circle

    Chapter 1, Section 1.2 Write the standard form of the equation of the circle with the given characteristics. 72. Center: (−2, −6); Solution point: (1, −10) Solution: given: Center: (−2, −6); => h=-2, k=-6 => then (x - (-2))^2 + (y - (-6))^2 = r^2 (x +2)^2 + (y +6)^2 = r^2... I then use...
  2. N

    Write the Standard Form of the Equation for this Circle

    Chapter 1, Section 1.2 Write the standard form of the equation of the circle with the given characteristics. 74. Endpoints of a diameter: (11, −5), (3, 15) I want to know if the following steps are correct for me to answer the above question. Steps: 1. Find the distance between the points...
  3. M

    MHB Express X in terms of L1, L2 and L3. Circle within Circle.

    I do not know if there is a solution to the problem. With the aid of CAD Software, I get X = 3.5 when L1=7.082, L2=0.684, L3=0.876. I don't know how accurate this is. My intuition says there should be a expression for X but I am yet to solve it. Any ideas?
  4. O

    MHB Finding the Center of a Circle

    Hello! I have an application where I need to find the center of a circle where I am having trouble coming up with a simple way to do this. The diameter of the circle is known and i want to be able to determine the location of it where only a portion of the circle is known. (see the image...
  5. karush

    MHB ASVAB circle and inscribed rectangle area problem

    Rectangle ABCD is inscribed in the circle shown. If the length of side $\overline{AB}$ is 5 and the length of side $\overline{BC}$ is 12 what is the area of the shaded region? $a.\ 40.8\quad b.\ 53.1\quad c\ 72.7\quad d \ 78.5\quad e\ 81.7$ well to start with the common triangle of 12 5...
  6. Ale_Rodo

    Lorentz force acting upon an electron moving in a circle

    So as the summary suggests, I am studying Electromagnetism, magnetic properties of matter and Magnetization vector in particular. As a first example and to introduce the Magnetization vector (M), my textbook shows a ferromagnetic substance in a uniform magnetic field (B). Then, every atom of...
  7. M

    Digital Filters: why is sampling frequency equal to 2*pi unit circle

    Hi, I was working through a filter design problem and got stuck on a concept. Scenario: Let us say we have the following pulse transfer function and the sampling frequency is ## f_s = 50 \text{Hz} ##. G(z) = \frac{1}{3} \left( 1 + z^{-1} + z^{-2} \right) The zeros of the transfer function...
  8. H

    Where Did I Go Wrong in Calculating Net Current Through a Circle?

    Here's what I did: ∮ B * dl ＝μ0 ＊ I ∮ AR * 2π*R ＝μ0 ＊ I ∮ 2π*AR^2 / μ0 = I ∮ 2π*AR^3 / 3μ0 = I Where did I do wrong?
  9. A

    MHB Circle a equation in a straight line

    how to circle a equation in a straight line y= 2x-1 y = x^2+ 2 x –y = 1 x = y + 5 y = – x^2 + 5 pls help me
  10. H

    MHB Calculate the radius of the circle

    A, B and C are points on a circle with center O. Angle ABC = $75°$ . The area of the shaded segment is $200cm^2$ . Calculate the radius of the circle. Answer correct to $3$ significant figures.
  11. M

    MHB Draw Angles & Find Values in Unit Circle

    Hey! :giggle: Make a drawing for each of the values of the angle below indicating the angle at the unit circle (in other words: $\text{exp} (i \phi )$) and its sine, show cosine, tangent and cotangent. Give these four values explicitly in every case (you are allowed to use elementary...
  12. L

    Unable to find the intersection between a circle and ellipse

    Given: x^2+xy+y^2=18 x^2+y^2=12 Attempt: (x^2+y^2)+xy=18 12+xy=18 xy=6 y^2=12-x^2 (12)+xy=18 xy=6 Attempt 2: xy=6 x=y/6 y^2/36+(y/6)y+y^2=18 43/36y^2=18 y ≠ root(6) <- should be the answer Edit: Just realized you can't plug the modified equation back into its original self I plugged y=6/x...
  13. M

    MHB What is the formula for finding the area of a circle?

    My Effort: Circumference = pi•d 10 •pi = pi•d 10•pi/pi = d 10 = d, where d is the diameter of the circle. Area = pi•r^2, where r is the radius of the circle. Diameter = 2 times the radius. 10pi = 2r 10pi/2 = r 5pi = r A = pi•r^2 A = pi(5pi)^2 A = 25•pi^3, which makes no sense. Only...
  14. Gh778

    B Energy to increase the radius of a circle composed of several disks

    Hi, I take a big number of disks to composed a circle of a radius of 1 m, the blue curved line is in fact several very small disks: I take a big number of disks to simplify the calculations, and I take the size of the disks very small in comparison of the radius of the circle. The center A1 of...
  15. A

    Determining the radius of a concentric circle.

    A concentric cirlce has two circles with the same center, but a different radii. We are given a pie with radius ##r##. A circular cut is made at radius ##r## such that the area of the inner circle is ##1/2## the area of the pie. We know that the formula to calculating the area of a circle is...
  16. S

    B Why does the radius of a unit circle need to be 1?

    Why is it that the radius of the unit circle is 1?
  17. AN630078

    Trigonometry: finding an angle, area and length of sector of a circle

    1. Using the formula for the arc length; s= θr I have endeavoured to find the angle AOB sine both the arc length and radius are known; 11= θ*8 θ=11/8=1.375 rad I actually do not think that this can be correct as it seem to simplistic a response. Have I misinterpreted the question or used the...
  18. SSequence

    B Points on a Circle (moving through an obstacle)

    This is a problem in two dimensions. Consider an obstacle (in the form of a line segment) placed at ##x=1##, ##0 \leq y \leq 1##. Now consider the circle of radius ##1## and center at ##(x_0,1)## [initially at time ##t=0##] moving towards right with a combination of: (i) Angular speed of ##2...
  19. burian

    How can I find the velocity of a point on a disk rotating in a disc?

    From a freebody analysis I got, $$ \vec{r} \times \vec{F} = |r| |F| \sin( 90 - \theta) = (R-r) mg \cos \theta$$ and, this is equal to $$ I \alpha_1$$ where the alpha_1 is the angular acceleration of center of mass of small circle around big one, $$ I \alpha = (R-r) mg \cos \theta$$ Now...
  20. LCSphysicist

    Analyzing Net Forces and Equations in a Driven Mass on a Circular Path System

    What you think about this system:? F*cosw - k(x1+x2) - k(x1-x2) = mx1'' -k(x1+x2) - k(x1-x2) = mx2''
  21. R

    Intersection of a circle and a parabola

    We have a circle (x^2 + y^2=2) and a parabola (x^2=y). We put x^2 = y in the circle equation and we get y^+y-2=0. We get two values of y as y=1 and y=-2. Y=1 gives us two intersection point i.e (1,1) and (-1,1). But y=-2 neither it lie on the circle nor on the parabola. The discriminant of the...
  22. A

    B Area of a Circle in an Electron's Hydrogen Atom

    My textbook says "A is the area of the circle enclosed by the current" (produced by an electron in a hydrogen atom), A = ##\pi r^2 \sin(\theta)^2##. I don't understand where the ##\sin(\theta)^2## comes from.
  23. anemone

    MHB Find the diameter of one circle

    Five identical semicircles are arranged as shown. Find the diameter of one circle. \draw (0,0) -- (16.5, 0); \begin{scope} \clip (0,0) rectangle (4.5,4.5); \draw (2.25,0) circle(2.25); \draw (0,0) -- (4.5,0); \end{scope} \begin{scope} \clip (6,0) rectangle (10.5,4.5); \draw...
  24. R

    Does a square shaped line may have a circle shaped Gauss' surface

    Summary:: For finding the electric field at P in the photo below, may I select a gaussian surface circular? [Mentor Note -- thread moved to the schoolwork forums, so no Homework Template is shown]
  25. S

    Kinematics problem — Calculating displacements around a circle

  26. P

    MHB Find the points of intersection of a line and a circle

    How do I algebraically prove how many times the line y=-5 intersects the circle (x-3)^2 + (y+2)^2 =25?
  27. Frigus

    What causes tension in a rope moving in a vertical circle?

    I don't get how ball moves in a vertical circle,we say tension provides centripetal force to the ball, i have posted a image in this post and in I which I have shown that their is a downward force mg and upward velocity,in this what will cause tension in the rope...from non inertial frame it is...
  28. jisbon

    How Do Symmetrical Charges Affect Net Electric Field in a Circle?

    In this case, I know there won't be any net efield in the x direction because it cancels out with each other. The problem is dealing with the y axis. Am I supposed to presume an angle for each of them or what should I do instead? Thanks
  29. A

    I Involuted planes within a circle, Klingelnberg-Palloid involute

    So this "seemingly simple" geometry and idea caught my attention. See the video in the link from 9:00 minute They talk about a specially designed nuclear fuel canister/bundle, now there is this geometry where they have a cylinder with smaller diameter and then a cylinder with a larger...
  30. T

    MHB The smallest circle that two parts of a semi-circle can fit into?

    So, true story: I made a large circular tortilla. Ate half of it. Then decided to put the rest into the fridge on a smaller plate.I raised the knife to cut the remaining semi-circle in two, and then went : "Hmmmmmmmm...". Anyway, it's in the fridge now with an approximate solution, but I'm...
  31. Monoxdifly

    MHB [ASK] Equation of a Circle in the First Quadrant

    The center of circle L is located in the first quadrant and lays on the line y = 2x. If the circle L touches the Y-axis at (0,6), the equation of circle L is ... a. x^2+y^2-3x-6y=0 b. x^2+y^2-12x-6y=0 c. x^2+y^2+6x+12y-108=0 d. x^2+y^2+12x+6y-72=0 e. x^2+y^2-6x-12y+36=0 Since the center (a, b)...
  32. Monoxdifly

    MHB [ASK] A Circle Which Touches the X-Axis at 1 Point

    The circle x^2+y^2+px+8y+9=0 touches the X-axis at one point. The center of that circle is ... a. (3, -4) b. (6, -4) c. (6, -8) d. (-6, -4) e. (-6, -8) I already eliminated option c and e since based on the coefficient of y in the equation, the ordinate of the center must be -4. However, I...
  33. R

    Line integral around a circle centered at the origin

    Hi everyone, I am confused in this question. First I solved it by noticing that the gradient of the function will be zero (without substitution the hit) I got that it's a conservative field so the integral should be zero since it's closed path. Then I solved it by the hit and convert it as any...
  34. karush

    MHB 2.1.312 AP Calculus Exam Int of half circle

    The function f is defined by $$f(x)=\sqrt{25-x^2},\quad -5\le x \le 5$$ (a) Find $f'(x)$ apply chain rule $$ \dfrac{d}{dx}(25-x^2)^{1/2} =\dfrac{1}{2}(25-x^2)^{-1/2}2x =-\frac{x}{\sqrt{25-x^2}}$$ (b) Write an equation for the tangent line to the graph of f at $x=-3$...
  35. J

    B Is this true? The area of a circle can be approximated by a polygon

    Hello everyone! I have been looking for a general equation for any regular polygon and I have arrived at this equation: $$\frac{nx^{2}}{4}tan(90-\frac{180}{n})$$ Where x is the side length and n the number of sides. So I thought to myself "if the number of sides is increased as to almost look...
  36. A

    Angular acceleration and velocity in a circle

    33.33 RPM (1min/60second)(6.28 radian/1 revolution) = 3.45 rad/s linear velocity=3.45 rad/s * 0.1524m=0.526 m/s linear acceleration= (0.526 m/s)^2 /(0.1524m)=1.814 m/s^2 1.814 m/s^2=(0.1524m)(rotational acceleration) rotational acceleration=11.9 rad/s^2 ω1= (3.45 rad/s)+(11.9rad/s^2)(10s)...
  37. J

    Collapse and deformation of a circle (tube)

    Hello: I am looking for a formula that can help me determine the collapse and deformation strengths of plastic tubing. I have been scouring the internet for this information and i have yet to find a satsifactory formula. I have found a formula that seems pretty wide spread ~ however it gives me...
  38. Mathman2013

    Optimization problem (Max area of a combined semi circle and a square)

    A figure is made from a semi circle and square. With the following dimensions, width = w, and length = l. Find the maximum area when the combined perimiter is 8 meter. I first try to construct the a function for the perimeter. 2*l + w + 22/7*w/2 = 8 - > l = 4 - (9*w)/7 Next I insert this...
  39. grandpa2390

    Find the area of this larger circle

    if it helps, the answer is supposed to be my colleagues and I can't figure out how to come to that answer. It's probably something simple. edit: I tried to solve it by inscribing an octagon, and then finding the distance from the center of the octagon to the side of the octagon. but I got 1 +...
  40. Monoxdifly

    MHB [ASK] Equation of a Circle

    find standard eqations of circles that have centers on 4x+3y=8 and are tangent to both the line x+y=-2 and 7x-y=-6 What I got is 4a=–4\pm3r\sqrt2 and b=4\pm r\sqrt2. Dunno how to continue from here.
  41. D

    What Does a Flat Disc Bounded by a Circle Mean in Electric Flux Calculations?

    Hi! My main problem is that I don't understand what the problem is telling me. What does it mean that the surface is a flast disc bounded by the circle? Is the Gauss surface the disc? Does that mean that inside the circle in the figure, there is a disc? Can you give me some guidance on how to...
  42. Like Tony Stark

    ##\vec v## and ##\vec a## expressions - motion on an off center circle

    Well, I tried decomposing velocity into its components on the radial and angular axis. But I have problems with the angles because in some parts of the trajectory the velocity is on the angular coordinate, but in other parts it isn't. I mean, I can't say ##V=V e_\theta## because it's not always...
  43. C

    Spinning a bucket of water in a vertical circle - Inertia

    Homework Statement: How or why does inertia caused the water in a bucket not to fall out when spinning in a vertical circle. Homework Equations: Is the bucket catching the water? I know Inertia is the resistance of any physical object to any change in its velocity.
  44. C

    Physics - Spinning a bucket of water in a vertical circle

    F = mac = (mv2 / r)
  45. AndresPB

    Electric Field from its Potential of a Half Circle along its Z axis

    So I figured out the potential is: dV = (1/(4*Pi*Epsilon_0))*[λ dl/sqrt(z^2+a^2)] . From that expression: We can figure out that since its half a ring we have to integrate from 0 to pi*a, so we would get: V = (1/(4*Pi*Epsilon_0))*[λ {pi*a]/sqrt(z^2+a^2)] In that expression: a = sqrt(x^2+y^2)...
  46. P

    I Circumference of a circle on a spherical surface

    I was just reading an intro text about GR, which considers the circumference of a circle on a sphere of radius R as an example of intrinsic curvature - the thought being that you know you're on a 2D curved surface because the circle's circumference will be less than ##2\pi r##. They draw a...
  47. D

    I How to know if a complex root is inside the unit circle

    Hi. I have been trying to calculate the real definite integral with limits 2π and 0 of ## 1/(k+sin2θ) ## To avoid the denominator becoming zero I know this means |k|> 1 Making the substitution ##z= e^{iθ}## eventually ends up giving me a quadratic equation in ##z^2## with 2 pairs of roots...
  48. karush

    MHB Finding the Equation of a Circle Passing Through Three Given Points

    find an equation of the circle passing through the given points 85 Given $(-1,3),\quad (6,2),\quad (-2,-4)$ since the radius is the same for all points set all cirlce eq equal to each other $(x_1-h)^2+(y_1-k)^2=(x_2-h)^2+(y_2-k)^2=(x_3-h)^2+(y_3-k)^2$ plug in values...
  49. Y

    MHB How Can I Find the Equation of the Dotted Tangent Line of a Circle?

    Dear all, Attached is a picture of a circle. The lower tangent line is y=0.5x. The center of the circle is M(4,7) while the point A is (3,6). I found the equation of the circle, it is: $(x-4)^{2}+(y-7)^{2}=20$ and I wish to find the dotted tangent line. I know that it is parallel to the...
  50. hilbert2

    Complex polynomial on the unit circle

    So, the values of polynomial ##p## on the complex unit circle can be written as ##\displaystyle p(e^{i\theta}) = a_0 + a_1 e^{i\theta} + a_2 e^{2i\theta} + \dots + a_n e^{ni\theta}##. (*) If I also write ##\displaystyle a_k = |a_k |e^{i\theta_k}##, then the complex phases of the RHS terms of...
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