# 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. ### Mathematica Black Hole shadow temperature profile

I am not sure if it is right to ask this question or not. Kindly let me know if it isn't. Actually I was going through the article Shadow Thermodynamics. I was trying to recreate the image fig 5 and 6 in mathematica. The idea is that we have to draw circles of radius ##r_s## given by equations...
2. ### B Question about change of variables

Hello everyone, I found a good proof for the area of a circle being ##{\pi}r^2## but I was recently working on my own proof and I used a change of variables and was wondering if I did it correctly and why a change of variables seems to work. I start with the equation of a circle ##r^2 = x^2 +...
3. ### Find the area of a segment of a circle using integration

Mentor note: Moved from a math technical section, so template is not present. I was asked to calculate the area of the smaller section enclosed by the circle x²+y²-6x-8y-35=0 and the x axis. I've tried to solve it with geometry, using the x-intercepts and the centre of the circle I drew a...

45. ### Distance from a point on a circle to an arbitrary axis

Hi all! In this assignment I have to formulate an equation for the shortest distance from a point on a circle perimeter to an arbitrary axis in a circle with angle theta. I included an image with the sketch. Anyone that can help?
46. ### Variance of a point chosen at random on the circumference of a circle

Hi, I was looking at this problem and just having a go at it. Question: Let us randomly generate points ##(x,y)## on the circumference of a circle (two dimensions). (a) What is ##\text{Var}(x)##? (b) What if you randomly generate points on the surface of a sphere instead? Attempt: In terms of...
47. ### MHB [ASK] A Line Intercepting A Circle

A circle whose center is (2, 1) intercepts a line whose equation is 3x + 4y + 5 = 0 at point A and B. If the length of AB = 8, then the equation of the circle is ... A. x^2+y^2-24x-2y-20=0 B. x^2+y^2-24x-2y-4=0 C. x^2+y^2-12x-2y-11=0 D. x^2+y^2-4x-2y+1=0 E. x^2+y^2-4x-2y+4=0 I don't know how to...
48. ### MHB -2.4.27 find center and radius of circle

Determine the graph of $x^2+y^2+6x+8y+9=0$ \$\begin{array}{rll} \textsf{rewrite} &(x^2+6x )+(y^2+8y)=-9\\ \textsf{complete square} &(x^2+6x+9)+(y^2+8y+16)=-9+9+16\\ \textsf{simplify equation} &(x+3)^2+(y+4)^2=16=4^2\\ \textsf{observation} &C(-3,-4), \quad R=4...
49. 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...
50. 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...