Questions tagged [circles]

For questions concerning circles. A circle is the locus of points in a plane that are at a fixed distance from a fixed point.

A circle is a shape in geometry, defined as the locus of points that have a fixed distance from a certain point, called the centre. The fixed distance from the centre of a circle to any of its points is called the radius. The length of the set of points is called the circumference, and for Euclidean space is related to the length of the radius by:


Similarly, in Euclidean space the area enclosed by a circle is given by:


Because of their radial symmetry and structure, circles have a large number of desirable properties. These include:

  • The circle is the shape with the largest area for a given length of perimeter.
  • The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle.
  • All circles are similar.
    • A circle's circumference and radius are proportional.
    • The area enclosed and the square of its radius are proportional.
  • The circle that is centred at the origin with radius 1 is called the unit circle.
    • Thought of as a great circle of the unit sphere, it becomes the Riemannian circle. Through any three points, not all on the same line, there lies a unique circle.
    • In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points.

There are many more properties of circles, see the following source for more information:

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Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers

An exam for high school students had the following problem: Let the point $E$ be the midpoint of the line segment $AD$ on the square $ABCD$. Then let a circle be determined by the points $E$, $B$ and $C$ as shown on the diagram. Which of the…
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Why can a Venn diagram for $4+$ sets not be constructed using circles?

This page gives a few examples of Venn diagrams for $4$ sets. Some examples: Thinking about it for a little, it is impossible to partition the plane into the $16$ segments required for a complete $4$-set Venn diagram using only circles as we could…
Larry Wang
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How many sides does a circle have?

My son is in 2nd grade. His math teacher gave the class a quiz, and one question was this: If a triangle has 3 sides, and a rectangle has 4 sides, how many sides does a circle have? My first reaction was "0" or "undefined". But my son wrote…
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Why is the derivative of a circle's area its perimeter (and similarly for spheres)?

When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle. Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is differentiated with respect to $r$, we get $4 \pi…
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How do you find the center of a circle with a pencil and a book?

Given a circle on a paper, and a pencil and a book. Can you find the center of the circle with the pencil and the book?
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The "pepperoni pizza problem"

This problem arose in a different context at work, but I have translated it to pizza. Suppose you have a circular pizza of radius $R$. Upon this disc, $n$ pepperoni will be distributed completely randomly. All pepperoni have the same radius $r$. A…
Ben S.
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A goat tied to a corner of a rectangle

A goat is tied to an external corner of a rectangular shed measuring 4 m by 6 m. If the goat’s rope is 8 m long, what is the total area, in square meters, in which the goat can graze? Well, it seems like the goat can turn a full circle of radius…
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Why is a circle in a plane surrounded by 6 other circles?

When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other numbers? I'm looking for an answer deeper than "there are…
John Smith
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Why is a circle 1-dimensional?

In the textbook I am reading, it says a dimension is the number of independent parameters needed to specify a point. In order to make a circle, you need two points to specify the $x$ and $y$ position of a circle, but apparently a circle can be…
mr eyeglasses
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Why is $\pi $ equal to $3.14159...$?

Wait before you dismiss this as a crank question :) A friend of mine teaches school kids, and the book she uses states something to the following effect: If you divide the circumference of any circle by its diameter, you get the same number, and…
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What is the size of each side of the square?

The diagram shows 12 small circles of radius 1 and a large circle, inside a square. Each side of the square is a tangent to the large circle and four of the small circles. Each small circle touches two other circles. What is the length of each side…
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Geometry problem involving infinite number of circles

What is the sum of the areas of the grey circles? I have not made any progress so far.
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How can I find the points at which two circles intersect?

Given the radius and $x,y$ coordinates of the center point of two circles how can I calculate their points of intersection if they have any?
Joe Elder
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Do circles divide the plane into more regions than lines?

In this post it is mentioned that $n$ straight lines can divide the plane into a maximum number of $(n^{2}+n+2)/2$ different regions. What happens if we use circles instead of lines? That is, what is the maximum number of regions into which n…
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A conjecture involving prime numbers and circles

Given the series of prime numbers greater than $9$, we organize them in four rows, according to their last digit ($1,3,7$ or $9$). The column in which they are displayed is the ten to which they belong, as illustrated in the following scheme. My…
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