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. S

    B Calculate distance between ends of a circle segment

    I was playing with a small piece of rope I had on the table, moved it to a curve - circle segment - and figured I wanted to calculate the distance between ends of the curve - and failed. Now I'm confused and embarrassed. Here is what I did: The rope has length L. I put the ends of the rope on a...
  2. paulb203

    B Understanding why πr^2 works for different area calculations

    I know there are videos etc explaining why but I thought I would try to find a way to understand this myself. Imagine a version of πr^2, but instead of being for the area of a circle, it’s for the area of a square. Call it sqi ar ^2 sqi = the ratio of the ‘diameter’ of the square to the...
  3. D

    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...
  4. C

    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 +...
  5. nafisanazlee

    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...
  6. S

    Line Integral of circle in counterclockwise direction

    My attempt: Let ##x=a \cos \theta## and ##y=a \sin \theta## $$\int_{L} xy^2 dx-x^2ydy$$ $$=\int_{0}^{2\pi} \left( (a\cos \theta)(a\sin \theta)^2 (-a\sin \theta)-(a\cos \theta)^2 (a \sin \theta)(a\cos \theta)\right) d\theta$$ $$=-a^4 \int_{0}^{2\pi}\left( \sin^3 \theta \cos \theta+\cos^3 \theta...
  7. J

    Derivation of angular velocity using the unit circle

    $$ \ ω = \frac{Δα}{Δt} \ $$
  8. DaveC426913

    B Inscribing a circle in an Oblique Square (Drafting)

    This comes up in a drafting and illustration contexts. It's a mix of 2D and 3D geometry. Since I was about twelve and first learning to draw mag wheels on racecars, I've been inscribing circles inside squares. (Not mine. Stolen off Google) I noticed right away that it is not as simple as it...
  9. P

    I Trajectory collision calculation

    Hello, I ask you for your aid in the solution of the following problem. Please see the attached illustration. Two objects (red and blue) are moving in the vicinity of each other. The red object is moving along a closed circle and the blue object is moving along a line. Our objective is to...
  10. W

    How do I remove this $#@ circle thing from my Android phone screen?

    Hi all, How fo I remove this circular object on the top right , the circle with the 4 square-ish objects?
  11. S

    Radii of stacked circles inside the graph of y = |x|^1.5

    (a) The hint from question is to used geometrical argument. From the graph, I can see ##r_1+r_2=c_2-c_1## but I doubt it will be usefule since the limit is ##\frac{r_2}{r_1} \rightarrow 1##, not in term of ##c##. I also tried to calculate the limit directly (not using geometrical argument at...
  12. notgoodatphysics

    B How Does Setting x=a Identify Minima or Maxima in Circular Equations?

    We have the equation for a circle, and its derivative: $$(y-a)^2 + (x-a)^2 = r^2$$ $$\frac{dy}{dx} = \frac{a-x}{(r^{2}-(x-a)^2)^{1/2}} = 0$$ So ##x = a## then he subs it into the original equation to get the max/min. Why does ##x = a## give the points of minima/ maxima if we didn’t know it...
  13. Jiketz

    Geometry: prove that point M is touched by 4 circles

  14. T

    I Is the Ratio of Circumference to Radius of a Circle Always Irrational?

    I take a string. It has a length, and that length is a real, rational number (let's call it an integer) I wrap it into a circle. Is it safe to say the radius is represented by an irrational number, too? Or is the "magic" (I would prefer not to use that word, but I go with it) of that number...
  15. D

    B Would light travel in a circle at the edge of the observable Universe?

    Is there sufficient mass within the observable universe’s volume to form a black hole event horizon around the observable universe and, if yes would light fired tangentially at the edge of our observable universe ever loop back around in a circle or spiral inwards?
  16. T

    What are the drawbacks to "annular wing"/"super circle" aircraft?

    By annular wing I mean something what is seen in this video. I'm fascinated by this design because it seems to lend itself to fluid entrainment and a greater lift profile for a smaller radius of wing. They are said to be quite stable.
  17. B

    Comp Sci Draw Longitude Lines of a Sphere on a Circle

    I'm trying to draw a 2D globe
  18. Y

    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...
  19. brotherbobby

    Finding Area of Shaded Segment in Circle Using Calculus

    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 Pythagorean...
  20. PainterGuy

    Medical How does human eye decode different geometric shapes such as circle?

    Hi, How does a human eye classify any shape as a circle, square, triangle etc.? Let's focus on a circular shape. Suppose we have a circle drawn in white on a black surface. The light falls on the retinal cells. I think the light falling on the retina will constitute a circular shape as well...
  21. N

    I Lens angle of coverage/image circle

    Hello, I was wondering what parameters determine the angle of coverage/size of the image circle of a lens. For example, for a fixed focal length, aperture and flange focal distance of a camera, what can a lens designer do to change the size of the image circle. I'd also like to know how the...
  22. V

    I Area of a Circle: Solving the Equation

    i can write the equation of circle easy enough, x^2+(y-r)^2=r^2. i get A=r^2/2 * asin((y-r)/r) + (y-r)/2 * sqrt(r^2 - (y-r)^2) through integration (using change of variable). Letting u = (y-r) and u^2=(y-r)^2, du= dy. Here's the rub... it's not right... :-) Appreciate and thanks in...
  23. chwala

    Solve the problem that involves a sector of a circle

    Wawawawawawawawa part (a) despite being easy had me running to and fro for some time!:cool: Anyway; Here we have; ##\dfrac{1}{2}r^2θ=\sin θ+\dfrac{3}{2}\sin θ## ##2θ=2.5\sin θ## the required result follows... I would appreciate an alternative approach to this... For part (b), we have...
  24. S

    Find how many points on a circle have an integer distance from other points

    Distance between point (-4, 5) and point on circle: $$d=\sqrt{(x+4)^2+(y-5)^2}$$ $$=\sqrt{x^2+8x+16+y^2-10y+25}$$ Then substitute ##y^2## from equation of circle: $$d=\sqrt{x^2+8x+16-x^2+4x-6y+12-10y+25}$$ $$=\sqrt{12x-16y+53}$$ After this, I need to try the points one by one to check whether...
  25. Y

    Heat circle -- Calculating the air flow needed across a condenser

    Hi! I need to calculate what air flow needs to flow across the condenser so that the heat emitted from the condenser is directed to the air. dT=10÷15 [C] COP =~3 According to the manufacturer's data the fan airflow is 0.67 [m^3/sec] and i need to check if i can use it. thanks
  26. chwala

    Solve the attached problem that involves circle and tangent

    Find Mark scheme here; Find my approach here...more less the same with ms...if other methods are there kindly share... part a (i) My approach is as follows; ##x^2+y^2-10x-14y+64=0 ##can also be expressed as ##(x-5)^2+(y-7)^2=10## The tangent line has the equation, ##y=mx+2## therefore it...
  27. L

    I How Fast Can a Magnet Train Go in a Circular Tunnel?

    I was wondering, if the magnet train is propelled by putting same polar magnet face each other at an angle and its natural magnetic property same pole repel, that's how it'spushing the train faster and faster...now what if we create a circular tunnel and have the train push itself and repel...
  28. chwala

    Find the area of the shaded region in the inscribed circle on square

    Find the solution here; Find my approach below; In my working i have; ##A_{minor sector}##=##\frac {128.1^0}{360^0}×π×5×5=27.947cm^2## ##A_{triangle}##=##\frac {1}{2}####×5×5×sin 128.1^0=9.8366cm^2## ##A_3##=##\frac {90^0}{360^0}####×π×10×10##=##78.53cm^2## ##A_{major...
  29. chwala

    Find the radius of the smaller circle in the tangent problem

    Find the question here and the solution i.e number 10 indicated as ##6-4\sqrt{2}##, I am getting a different solution, my approach is as follows. I made use of pythagoras theorem for the three right angle triangles as follows, Let radius of the smaller circle be equal to ##c## and distance...
  30. dom_quixote

    B Strange Relationships of the Circle

    PHOTOGRAPHIC REDUCTION OR ENLARGEMENT The proportions of a circle never change. But... Question: If a circle is always a circle, then how is it possible that the S/L and {[S/L]/R} ratios of a circle can vary as a function of the radius size?
  31. DaalChawal

    MHB What is the internal tangent circle problem for three given circles?

    Help in circle touching internally all these three circles.
  32. Remixex

    Triangles inside a circle to represent raypaths inside an ideal Earth

    I have managed to get some of the required distances and angles. I have the distance ##a##, the velocity inside the mantle, the total radius of the Earth ##R_t## as well as mantle and core radii. I have also figured out the angle of incidence, however I cannot get the refracted angle with the...
  33. U

    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...
  34. Al-Layth

    Circular Motion Problem -- Ball on a String Spinning in a Vertical Circle

    #F= m\frac{v^2}{r} = mw^{2}r# #m=5# #r=0.9# #F= 5\frac{v^2}{0.9} = (0.9)5w^{2}# #5\frac{v^2}{0.9} = (0.9)5w^{2}# #\frac{v^2}{0.9} = (0.9)w^{2}# #v=0.9w# then I get stuck cause I have both unknowns in one equations (i bet it has something to do with the question’s use of “minimum” but I...
  35. Perchaddition

    B Can I use the circle circumference formula for a sphere?

    Trying to calculate a circumference of a sphere from a radius of 3.09 inches. Is 19.4 a correct answer? Just ran numbers in the first circumference calculator I found http://calcurator.org/circumference-calculator/. Can I use the same formula for a sphere? What can I say ...Geometry is not my...
  36. D

    One-parameter parametrization of a unit circle in R^n

    I tried to looking at lower-dimensional cases: For ##n=2## we have $$(x(t),y(t))=(cos(t),sin(t))$$ For ##n=3## we define two orthogonal unit vectors ##\vec{a}## and ##\vec{b}## that are orthogonal to ##\vec{u}##, leading to $$(x(t),y(t),z(t))=(cos(t)(a_1,a_2,a_3)+sin(t)(b_1,b_2,b_3))$$ It seems...
  37. J

    I Proof of average height of half circle

    I need proof how find average height of half circle? Lets say pressure distribution is half circle with Pmax = radius,I must find average/resultant pressure..
  38. DaalChawal

    MHB Calculating rates of two points moving along a circle

    Question 3
  39. Nikkki

    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...
  40. Leo Liu

    Notation in fluid dynamics: A circle with a horizontal bar inside

    I came across this notation when reading the wiki article on Venturi effect, which baffled me. Can someone please tell me its meaning?
  41. applesoranges

    Mass swinging in a horizontal circle on 2 strings

    I have already solved this problem, just would like to double check something with you conceptually. I've got a negative result for the tension in the lower cord. Intuitively I think it is right, because the lower cord does not support the ball in its opposing the force of gravity. It actually...
  42. G

    I Encircled energy for different aperture shapes (circle, triangle, square)

    Hi all, I have a system whereby, there are different aperture shapes which are: circle, triangle, square e.t.c. this apertures are all 300um in diameter. I will like to know if the encircled energy calculated for the different apertures after diffraction will be different due to different...
  43. Poetria

    Parametrised Circle Trajectory: Particle in x≤0 Half-Plane

    The trajectory of a particle is a semi-circle contained in the x≤0 half-plane. Well, this is somewhat weird. I have come across examples with x(t)=cos(t), y(t)=sin(t) and not the other way round. By the way, my answer is wrong but I don't know why. This is probably silly. :(
  44. B

    I Equation to graph a sine wave that acts like a point on a unit circle

    I need an equation to graph a sine wave that act like a unit circle but only positive numbers. so I need it to be 0 at 0, A at 90 , 0 at 180, A at 270, 0 at 360, and A at 450 and so on and so on... Now I know sin(0) is 0 in degrees and sin(90) 1 and I know if you Square a number is...
  45. Mathman2013

    Find the coordinates of a point C from the given line, point and circle

    Let the point P(2,8) be a point in xy-plane and line m: y = -0.75*x+3.25 be a line in the xy-plan. The distance from a point P to a point B is 7 unites. Where the x coordinate of B is negative. Find the acute angle between PB and m. To find B I then construct a circle of radius 7 with center...
  46. S

    Force exerted by rod on a mass moving in vertical circle

    (A) and (B) are obviously wrong but I think both (C) and (D) are correct. At the top, the forces acting on the mass are tension and weight, both directed downwards so the equation of motion will be: $$\text{Tension}+\text{Weight}=m.a$$ $$\text{Tension}=m.a-\text{Weight}$$ Based on that...
  47. Slipjoints

    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?
  48. M

    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...
  49. Monoxdifly

    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...
  50. karush

    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...
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