# 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.

359 questions
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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 ...
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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?
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### Parametric Equation of a Circle in 3D Space?

So, my dilemma here is... I have an axis. This axis is given to me in the format of the slope of the axis in the x,y and z axes. I need to come up with a parametric equation of a circle. This circle ...
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### How to determine the arc length of ellipse?

I want to determine the length of an arc from the ellipse in the picture below: How can I determine the length of $d$?
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### Area of intersection between two circles [duplicate]

Suppose you have 2 circles that intersect each other in such a way that each circle passes through the other's center. What is the area between the circle(or common area) i.e. area between the centres ...
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### What is the probability that the center of the circle is contained within the triangle?

Consider the triangle formed by randomly distributing three points on a circle. What is the probability of the center of the circle be contained within the triangle?
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### Determine Circle of Intersection of Plane and Sphere

How can the equation of a circle be determined from the equations of a sphere and a plane which intersect to form the circle? At a minimum, how can the radius and center of the circle be determined? ...
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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 ...
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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 $4$-...
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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 ...
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### Calculus proof for the area of a circle

I was looking for proofs using Calculus for the area of a circle and come across this one $$\int 2 \pi r \, dr = 2\pi \frac {r^2}{2} = \pi r^2$$ and it struck me as being particularly easy. The only ...
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### Numbers of circles around a circle

"When you draw a circle in a plane of radius $1$ you can perfectly surround it with $6$ other circles of the same radius." BUT when you draw a circle in a plane of radius $1$ and try to perfectly ...
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### How is the value of $\pi$ ( Pi ) actually calculated?

When I was a child I was taught $\pi$ (Circumference/Diameter) is an irrational number and can be approximated to $22/7$ but $= 3.(142857)(\ldots)$. But where does this value comes from? In ...
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### How to find an end point of an arc given another end point, radius, and arc direction?

Given an arbitrary arc, where you know the following values: end point (x1,y1), radius (r) and arc direction (e.g. clockwise or counterclockwise from start to end), how can I calculate the other ...
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### How to find the smallest enclosing circle over a set of circles?

I have a set of circles on a plane, each defined by their X, Y and R parameters. I need to find the smallest enclosing circle over all of them. Essentially, this is a variation of this question, ...
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### Why is $\pi r^2$ the surface of a circle

Why is $\pi r^2$ the surface of a circle? I have learned this formula ages ago and I'm just using it like most people do, but I don't think I truly understand how circles work until I understand why ...
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### Calculate the area of the crescent

I found this problem on a thread on Stack overflow where it was posted as "job interview question". Unfortunately I cannot find the question. But I saved the picture and just cannot figure it out. ...
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### How to find the equation of a line tangent a circle and a given point outside of the circle

I am given the equation of a circle: $(x + 2)^2 + (y + 7)^2 = 25$. The radius is $5$. Center of the circle: $(-2, -7)$. Two lines tangent to this circle pass through point $(4, -3)$, which is outside ...
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### Find the differential equation of all circles of radius a [closed]

Can someone please post a detailed step-by-step procedure. Given the circle with a radius a, what is the differential equation of the circle.
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### Circle areas on squared grid

There is a circle. On 9 equal squares. Every square has some value assigned to it. Every square gets weight, depending of what percentage of it is circle (area-wise). I need to find circle radius, ...
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### Two Circles and Tangents from Their Centers Problem

Let $\Gamma_1$ and $\Gamma_2$ be two non overlapping circles with centers $O_1$ and $O_2$ respectively. From $O_1$, draw the two tangents to $\Gamma_2$ and let them intersect $\Gamma_1$ at points $A$ ...
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### A chain of six circles associated with a cyclic hexagon

I found the problem some months ago. But I never have been a proof. So I am looking for a proof. The problem as following: Let $ABCDEF$ be a cyclic hexagon. Let $(C_{AD})$, $(C_{BE})$, $(C_{CF})$ ...
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### How calculate the shaded area in this picture?

Let the centers of four circles with the radius $R=a$ be on 4 vertexs a square with edge size $a$. How calculate the shaded area in this picture?
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### How to draw ellipse and circle tangent to each other?

The circle $c$ is given as are the points $A$ and $B$, which are ellipse's foci. Now I need to construct the ellipse that is tangent to the circle $c$ such that the points $A$ and $B$ are its foci. ...
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### Is it possible to build a circle with quadratic Bézier curves?

i'm searching for a curve type with a minimum of functionality and maximum of usability. I run into quadratic Bézier curves and i wonder, if its possible to draw a circle with it.
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### Find the approximate center of a circle passing through more than three points

Consider n point $(x_1,y_1), (x_2,y_2),\ldots, (x_n,y_n)$. For $n = 3$ it is easy to find the center of the circle passing through the three points. I wanted find the approximate center of the circle ...
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### Ill-known/original/interesting investigations on/applications of inversion (the geometric transform)

Inversion transform with center (or pole) $C$ and power $k^2$ is defined by: $$\tag{1}J_{C,k}:M \leftrightarrow M' \ \ \ \ \ \iff \ \ \ \ \ \ \ \vec{CM'}=\frac{k^2}{||\vec{CM}||^2} \ \vec{CM}$$ It ...
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### Probability distribution for the perimeter and area of triangle with fixed circumscribed radius

Given a circle with radius R = 1, I'm trying to find either the probability distribution function or the density function for the space of triangle, which is randomly selected on this circle. The same ...
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### Number of Squares in a circle

I a math question, that I hope someone can help me with. I have 342 Squares sized at 11 x 11 cm and need to calculate how to pack them in a circle and find out how large the circle must be to pack ...
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### Prove that the triangles $ABC$ and $AB^{'}C^{'}$ have the same incentre.

The question is as follows if $ABC$ is a triangle, with $AD$ as the internal angle bisector of $\angle A$ with $D$ at $BC$ and $B^{'}, C^{'}$ are reflection of points $B$ and $C$ in $AD$. Show that ...
Let $A\subset\Bbb C$ be a subset of the unit circle. Consider the following condition on $A$. Cond. There exists a sequence $\{a_i\}_{i=1}^\infty$ of complex numbers such that \sum_{n=1}^\infty ...