Circle Definition and 52 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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2. Coloring each k-th unit in a circle of n units

suppose you write, clockwise, n numbers (or "units", doesn't matter) in a circle. you then color, clockwise, each k-th number. you do this until you've colored all n numbers, or until you've reached an already colored number. let x be the number of colored numbers. i've figured that if...
3. Area of a segment of a circle

Problem Statement : To find the area of the shaded segment filled in red in the circle shown to the right. The region is marked by the points PQRP. Attempt 1 (without calculus): I mark some relevant lengths inside the circle, shown left. Clearly OS = 9 cm and SP = 12 cm using the...
4. I Understanding 3D circle parameterization

Hey I saw this post: https://www.physicsforums.com/threads/general-equation-of-a-circle-in-3d.123168/ trying to understand 3D circle parameterization. I saw the formula given by Hootenanny But I didn't understand it and would like some help. What are u and n and how can I find/calculate them...
5. I Equation for circle points in 3D

Hello, I am trying to solve a problem and I would like to ask for help. I have 3 points (A, B, C) in 3D space that are assumed to be on a circle. EXAMPLE 1 EXAMPLE 2 My goal is to create an algebraic formula to calculate the coordinates for 10 points on a circle composed of ABC points at...
6. 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...
7. B Surface area problem -- Area of the circle enclosed by the current produced by an electron in a 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.
8. 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...
9. 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...
10. Find this angle given the triangle's Orthocenter

##AD## is diameter, thus ##\angle ACD = \angle ABD = 90^\circ##. Also ##HBDC## is a parallelogram because ##HC||BD, HB||CD##. It seems useless and I don't know how to continue. Thank you in advance!
11. Angle between radius and acceleration

Homework Statement A particle is moving clockwise in a circle of radius 2.50m at a given instant of time. I have to find radiant and tangential acceleration and the speed of the particle. The acceleration vector is 15.0 m/s² and the angle between the radius and the acceleration vector is 30°...
12. Find the center of a circle given a tangent line & point

Homework Statement "Find the center and radius of the circle that passes through A(1,1) and is tangent to the line y=2x-3 at the point B(3,3). (Picture of the graph: https://imgur.com/a/0wAnqcU) Homework Equations Here's a link: https://imgur.com/a/y71Z9GY The Attempt at a Solution Soo, I've...
13. Points and lines

I have no idea what to do please help me. although i did this for the second equation, x/2h+y/2k=1 this represents an elipse first equation is circle
14. 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...
15. 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...
16. 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...
17. 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...
18. Finding dy/dx 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"...
19. How to find chords, intersections of chords on circle?

Homework Statement Homework Equations The 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?
20. 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...
21. 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...
22. 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...
23. B Stumped by the simplest geometry problem

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...
24. 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...
25. Circle Exercise Question

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...
26. 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...
27. 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)...
28. 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/CAacV2dJbUrywfXv9 2. The attempt at a solution It seems I found a solution for this exercise with the help of...
29. 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...
30. I Area of Appolonian Gasket

Here is my formula for the area of n layers of appolonian gasket(assuming no circles past the nth layer): $$πR^2 - (πR^2 - (\sum_{0}^{n} x_n*πr_{n}^2))$$ Here R is the radius of the outer circle, r is the radius of an inner circle, x is a function that represents the number of circles in a...
31. I Surface of revolution of a donuts

HELP I can't find the surface of revolution! By donuts I mean a circle that doesn't touch the axes (tore in french) y^2+(x-4)^2=2^2 is my function ( y^2+x^2=r^2) and the axe of rotation is y so y= sqrt(r^2-x^2) the formula I know : 2* pi (Integral from a to b (F(x)*sqrt( 1+ (f(x))^2))...
32. Parametric equation of a circle intersecting 3 points

Homework Statement Given the points P0 = (0,a), P1 = (b,0), P2 = (0,0), write the parametric equation of a circle that intersects the 3 points. Assume that b > a and both are positive. Homework Equations X = h + rcos(t) Y = k + rsin (t) r = √((x-h)2 + (y-k)2 Cos (t) = (x-h)/r Sin (t) =...
33. Friction and turning angle relation

Hi there, Suppose we had a car going in a circle. We know that the turning angle(angle between the movement direction and the wheel axis) and the friction are connected, because friction determines the centripetal force and it determines the radius of the circle that we make with our car. I need...
34. Find the center of the circle of curvature

Homework Statement For the curve with equation y={ x }^{ 2 } at the point (1, 1) find the curvature, the radius of curvature, the equation of the normal line, the center of the circle of curvature, and the circle of curvature. Homework Equations The Attempt at a Solution \kappa \left( 1...
35. I S^1 transition functions

I am confused about the procedure for finding the transition functions given an atlas. I understand the theory; it's applying it to real life examples where I have my problem. So for example, take S1 (the circle). I want to use 2 charts given by: U1 = {α: 0 < α < 2π} φ1 = (cos α, sin α) U2...
36. Why is there "weightlessness" on the top of a verticle circle?

i'm ashamed, that i never understand this, eventhough I'm studying quantum mechanics... so... why is there "weightlessness" on the top of a verticle circular motion? ie, if a plane if flying in verticle circles, why is there weightlessness while on the top of a circular path? i mean, if it's...
37. Integrals to Solve Area and Center of Mass of a Cut Circle

Homework Statement I am after finding the centroid of the remaining area (hatched) when a circle is cut by a line. I made a diagram in CAD that demonstrates the problem. The idea is that, starting from the bottom of the circle, a cut is taken leaving a remaining shape whose area and...
38. Area of circle using integration

Homework Statement http://postimage.org/][/PLAIN] [Broken] free picture upload 2. The attempt at a solution I want to go width times delta height. To do this I must describe width in terms of height. Here they used the Pythagorean theorem which is weird to me because I don't see a nice...
39. Circle/Sphere touching

Theoretically, if you had a perfect circle, and a perfectly flat surface, wouldn't only one atom touch at a time (assuming friction can't take away the perfect circle/flat surface)? Personally this doesn't sound right, but I can't think of why it wouldn't.
40. Why are circles infinitely smooth if they have degrees?

Because a triangle comes out to 180 degrees, and yet it can only have three sides. A circle has 360 degrees, but its number of "sides" are uncountable. Can someone explain this?
41. B Do curves, circles and spheres really exist?

Obviously, they exist as mathematical concepts, and those concepts are real, but in physical reality, everything is made up of subatomic particles and, if the theory is ever verified, strings. So if you try to construct a curve, circle or sphere, you are necessarily stacking a bunch of subatomic...
42. Calculate the distance travelled in a moving circumference?

First of all I don't mean a rotating circumference, but rather a translating one which is also rotating. Like this one: https://www.desmos.com/calculator/4vdymlzgpp just play the p button. I understand that the x component of the movement is: and Y= (where v is the radius, s is the angular...
43. [Euclidean Geometry] Kiselev's Plainimetry Question 242

Homework Statement Two lines passing through a point Μ are tangent to a circle at the points A and B. Through a point С taken on the smaller of the arcs AB, a third tangent is drawn up to its intersection points D and Ε with MA and MB respectively. Prove that (1) the perimeter of ▲DME, and (2)...
44. Minimum initial velocity

Homework Statement At which minimum velocity should you throw the ball horizontally if you are standing on a hemispherical rock of radius R so that it at no point touches the rock and lands at the minimum distance from the rock horizontally. Find the expression that solves for initial velocity...
45. Equation of family of circles

In my textbook, its given that the equation of family of circles touching a given circle S and line L is ##S+\lambda L=0## So to find the equation of family of circles touching line L at point P(p,q), can i use the same equation taking S to be a circle of radius zero and center at P? That is...
46. Conservation of Angular Momentum & Energy question

Question: A streetcar is freely coasting (no friction) around a large circular track. It is then switched to a small circular track. When coasting on the smaller circle its speed is: a) greater b) less c) unchanged Relevant Formulas: w = v/r KE = 1/2mv2 My teacher said the normal force from...
47. Vertical Circle question

Homework Statement The questions asks for the centripetal acceleration of the mass in the vertical circle shown in the picture. It also states the object has exactly enough velocity to maintain a vertical circle. Homework Equations F=ma, a=v^2/r The Attempt at a Solution I've tried solving...
48. Tangent to ellipse also tangent to circle

Homework Statement if the tangent at a point P("theta") on the ellipse 16 (x^2) + 11 (y^2) = 256 is also tangent to the circle (x^2) + (y^2) + 2(x) = 15 then ("theta") = ?? 2. The attempt at a solution {{{{ i have taken "theta" as "d" }}}} P [4 cos d , (16/(sqrt11)) sin d] equation of...
49. Circular Motion of a bucket

Homework Statement A 3.75 kg bucket pile of water is swung in a vertical circle. If the speed of the bucket at the top of the loop is 6.20 m/s, then the radius of the largest circle through which this pail could move without the water leaving the bottom of the pail would be what? m = 3.75 kg...
50. What is the minimum frequency to keep the mass moving?

Homework Statement A mass of 2.0 x 10^2 g is tied to a 1.6 m long string and spun in a vertical circle. What is the minimum frequency to keep the mass moving? Homework Equations Fc=m4(pie)rf^2 The Attempt at a Solution I know I have to use this equation and make it equal to another to find Fc...