Learn the Arc Length of a Circle Formula with Plainmath's Expert Help

Given a circle with diameter $d$, is there a way to find the length of the chord f that cuts off an arc of length $l$?

I am trying to understand the relationships between chords and arcs etc, and this concept came up - is there a general formula for it?

(And before I get shut down for this being a homework question, it is not. The concept came up in independent study.)

What is the formula to calculate the distance (arc length) between 2 points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ on the circumference of a circle of radius $r$ without knowing the angle $\mathrm{\theta <!--\; \theta \; -->}$ between them. I found that arc length can be calculated knowing $\mathrm{\theta <!--\; \theta \; -->}$. But I know only the $x,y,z$ co-ordinates of 2 points on the circumference of a circle. Please suggest.

Parametrize the contours of integration and write the integrals in terms of the parametrizations. Do not calculate them.
$\int <!--\; \int \; -->\frac{\stackrel{¯<!--\; ¯\; -->}{(}z)}{z^3}dz$
where
$\mathrm{\Gamma <!--\; \Gamma \; -->}$
is the arc of the circle of radius
$\sqrt{2}$
centered at the origin with initial point at 1+i and terminal point at 1-i that lies in the right half-plane and is transversed once.

I asked a question similar to this last night and felt that everyone was very helpful. I'm hoping to see the solution to this problem as I feel completely lost.

I am given three co-ordinate points of a circle O(Ox,Oy) as a center. Then two Points other points as A(Ax,Ay) & B(Bx,By). Now I have to find the arc length of that circle. Can you please help. Thanks in advance.

Consider the circle of radius r pictured below with central angle , measured in radians, and subtended arc of length s. Prove that the area of the shaded sector is $A=\frac{1}{2}{r}^2\mathrm{\theta <!--\; \theta \; -->}$.

Let's say the entire circle has area $B=\mathrm{\pi <!--\; \pi \; -->}{r}^2$. Using proportionality arguments, the area of a portion of a circle is then $\frac{\mathrm{\theta <!--\; \theta \; -->}}{2\mathrm{\pi <!--\; \pi \; -->}}B=\frac{\mathrm{\theta <!--\; \theta \; -->}}{2\mathrm{\pi <!--\; \pi \; -->}}\mathrm{\pi <!--\; \pi \; -->}{r}^2$ which reduces to $A=\frac{1}{2}{r}^2\mathrm{\theta <!--\; \theta \; -->}$.
Is that right? It seems too easy for a prove. I got this question from a online book which doesn't contain the answer, so I can only resort to stack exchange.

If I know the length of a chord of a circle and the length of the corresponding arc, but do not know the circle's radius, is there a formula by which I can calculate the length of the sagitta?

Find the arc-length of a circle with radius a.
From the equation of a circle, I found out the equation for the one quadrant, which is:
$y=\sqrt{a^2-<!--\; -\; -->x^2}$
I tried solving the problem, and here's what I did:
$A=4{\int <!--\; \int \; -->}_0^a\sqrt{1+\frac{x^2}{a^2-<!--\; -\; -->x^2}}dx$
$A=4{\int <!--\; \int \; -->}_0^a\frac{a}{\sqrt{a^2-<!--\; -\; -->x^2}}dx$
And this is where I need help. How can solve this integral now?

Let $n$ points be placed uniformly at random on the boundary of a circle of circumference 1.

These $n$ points divide the circle into $n$ arcs.

Let $Z_i$ for $1\le <!--\; \le \; -->i\le <!--\; \le \; -->n$ be the length of these arcs in some arbitrary order, and let $X$ be the number of $Z_i$ that are at least $\frac{1}{n}$.

What is $E[X]$ and $Var[X]$?

Any hints will be appreciated. Thanks..

(By the way this problem is exercise 8.12 from the book 'Probability and Computing' by Mitzenmacher and Upfal)

The length of an arc of a circle is 12 cm. The corresponding sector area is 108 cm${}^2$. Find the radius of the circle.
I have not attempted this question and do not understand how to solve this.

Today, I ran into a calculus class and the professor was calculating the perimeter of a circle using arc lenghts. Even though they seemed to be on the right track, there is a problem:

They are using sinx and cosx functions, which are actually defined using the perimeter of a circle, aren't they? So, in my opinion, that was not a real proof since they were using what they want to show.

Or am I missing something? Was that a real proof?

Edit: I don't know but most probably they defined sinx and cosx in the usual way, because it was a calculus class.

I am attempting to determine the locus of z for $\arg <!--\; \; -->(z^2+1)$. I do know that since $z^2+1=(z+i)(z-<!--\; -\; -->i)$, I could write $\arg <!--\; \; -->(z^2+1)=\arg <!--\; \; -->(z+i)+\arg <!--\; \; -->(z-<!--\; -\; -->i)$ but I don’t really know how to figure out the locus of the points from here. How should I approach this? Is there a relationship with the difference in two arguments (arc of a circle)? Thank you!

What angle is subtended at the center of a circle of radius 2 km by an arc of length 9 m?

Let
$f(z)={\displaystyle \frac{z(z-<!--\; -\; -->a)}{(z-<!--\; -\; -->b)(z-<!--\; -\; -->c)(z-<!--\; -\; -->d)}}$
be a complex rational function and let $S=\{z\in <!--\; \in \; -->\mathbb{C}:|z|=1\}$ be the unit circle.

My question:
For which (non-zero complex numbers) $a,b,c$ and $d$ there exist an arc $C$ of $S$ such that $f(C)\subseteq <!--\; \subseteq \; -->S$ ?

How to find the center of a circle with given an arbitrary arc. we only have the arc nothing else. Is there any known equation or way to complete the circle.

If the arc length and chord length between two points (two points on a circle that constitute a minor arc ) in a circle are known , find radius of the circle?

It is not possible for a part of any of three conic sections to be an arc of a circle.
It is asked to prove this theorem without using any notation or any modern form of symbolism whatsoever (algebra). But, to me, this seems to be quite a difficult task. How can this be done?

Suppose I'm given two points: $(x_1,y_1)$ and $(x_2,y_2)$ (which are real numbers) lying on the circumference of a circle with radius r and centred at the origin, how do I find the arc length between those two points (the arc with shorter length)?

I am trying to figure out what the best method is to go about finding this locus.
$\arg <!--\; \; -->{\displaystyle \frac{z-<!--\; -\; -->a}{z-<!--\; -\; -->b}}=\mathrm{\theta <!--\; \theta \; -->}$
I am aware that it must be part of an arc of a circle that passes through the points $a$ and $b$. The argument means that the vector $(z-<!--\; -\; -->a)$ leads the vector $(z-<!--\; -\; -->b)$ by $\mathrm{\theta <!--\; \theta \; -->}$ and so by the argument must lie on an arc of a circle due to the converse of 'angles in the same segment theorem'.

My question is, how do I know what the circle will look like, ie. the centre of the circle and the radius. If I don't need to know this, how will I know what the circle looks like.

Finally, if I changed the locus to
$\arg <!--\; \; -->{\displaystyle \frac{z-<!--\; -\; -->a}{z-<!--\; -\; -->b}}=-<!--\; -\; -->\mathrm{\theta <!--\; \theta \; -->}$
Would this just be the other part of the same circle as previously or a different circle?

Perhaps you could help by showing my how it would work with the question
$\arg <!--\; \; -->\frac{z-<!--\; -\; -->2j}{z+3}=\pi /3$