All Questions
96 questions
2
votes
4
answers
411
views
What is the intuition behind calculation of the area of circle?
I am learning about approximating the area of circle ($\pi r^2$) by the area of polygons ($\frac{1}{2}$ of number of sides * the distance between side and center * length of the side).
My question is ...
- 2,640
0
votes
0
answers
69
views
If I have x and y coordinates of a point along the arc, how do I convert that to a percentage of PI?
I am using Javascript to create shapes in canvas. I am creating an arc where you specify the start and end angle of the arc to show where along a circle the arc begins and ends. They are initially set ...
- 1
13
votes
1
answer
490
views
Covering a circle using rectangles
What is the maximum area that can be covered with $3$ rectangles inside
a radius $1$ circle?(i.e. maximum area $=\pi$) The rectangles can be any length and height you want, and can rotate and reflect.
...
- 193
3
votes
0
answers
150
views
Approximate $\pi$ using Gauss Circle Problem and Pick's Theorem.
Introduction
For any non-negative integer $k$, $x^2+y^2=5^k$ has $4(k+1)$ integer solutions (vertices).
For example, here are 3 circles for $k=1,2,3$. The blue dots represent the vertices (integer ...
- 2,113
11
votes
2
answers
483
views
Is there a question that contains no numbers except $1$, whose answer is $\pi/7$?
Is there a question that contains no numbers, except possibly $1$, whose answer is $\pi/7$ ?
There are plenty of questions with no numbers except $1$, whose answer is $\pi/n$ for small integer values ...
- 30k
3
votes
2
answers
571
views
Finding the formula for the circumference of a circle with sequences
Today I had an idea on how to find the formula for the circumference of a circle: $C = 2\pi r$, where $r$ is the radius of a circle. The idea is that we start with an equilateral triangle ($3$ sides) ...
- 195
4
votes
1
answer
206
views
Randomized algorithm to estimate $\pi$
I was looking for an algorithm to create a PI estimator, and I ran across this:
https://stackoverflow.com/questions/36659034/trying-to-create-a-pi-estimator-in-r
Briefly, the steps are:
...
- 85
4
votes
1
answer
178
views
Ratio between circumference and "radius" of a polygon
Given some polygon $P$ in two-dimensional Euclidean space, I want to define the radius of $P$ as the average of the radii of the smallest outer circle and the largest inner circle. An outer circle has ...
- 73
0
votes
0
answers
91
views
Prove that $\pi < \sqrt2 + \sqrt3$ [duplicate]
A square inscribed inside of a circle with radius $1$ must have a perimeter $4\sqrt2$. A regular hexagon circumscribed about that same circle has a perimeter $4\sqrt3$. Since we know the circle’s ...
- 939
0
votes
0
answers
75
views
Estimating Pi by Throwing Bread
I remember hearing about a story in which an Italian King (hundreds of years ago) drew a circle and randomly threw bread behind his shoulder, and calculated the percent of bread that landed inside the ...
- 3,437
0
votes
1
answer
73
views
is there any size of a circle where the area is an integer and the radius is an integer? [closed]
The formula for the area of a circle is pi times the radius squared. The radius is the diameter divided by 2. Imagine a line, like the axis, but instead, it doesn’t go past the edges. Now, the length ...
2
votes
0
answers
438
views
Area of a circle as sum of infinite squares
Take a quarter of a circle and divide it in squares as shown in the (very badly drawn) picture:
Every time you draw a new square take as much space as possible from the circle. This way every square ...
- 91
1
vote
2
answers
76
views
$\text{Angle}=\dfrac{\text{Arc length}}{\text{Radius}}$, a result or a definition?
There is a commonly used formula to find an angle
$$
\text{Angle}=\dfrac{\text{Arc length}}{\text{Radius}}.
$$
My question is whether this is a deduced formula or it is the very definition of an angle?...
- 546
1
vote
0
answers
51
views
Approximating $\pi$ with the help of a regular $k$-sided polygon
I'm reading "An Introduction to Computational Physics" by Tao Pang. In it, he writes the following. In general, if the side length of a regular inscribed $k$-sided polygon is
denoted as $l_k$...
0
votes
0
answers
70
views
Why can $-n\cdot \pi $ be changed to $n\cdot \pi $?
This task was just to solve this equation: $\cos2x=\cos4x$. I solved it correctly apart from one step. My book somehow changes $-n\cdot \pi $ to $n\cdot \pi $. How is it possible?
- 45
2
votes
2
answers
324
views
How is e to the hyperbola what pi is to the circle?
I'm trying to find a nice similarity between e and pi and I thought of conic sections. If you have a circle then the perpendicular conic section to that is a hyperbola. So this seems pretty similar. I ...
- 619
0
votes
0
answers
80
views
How is value of one radian and pi same for all circles?
For any two segments of $2$ different circles, how do we know that the angle made (we call it one radian for all circles) by arc length (equal to length of radius) to radius would be same as that of ...
2
votes
1
answer
60
views
What's the connection between hypotenuse-squared being integers and simple fractions of $\pi$ or whole circles?
I noticed something interesting when I think of Pythagoras theorem as addition of two integers to get another integer. For example, if the hypotenuse-squared is equal to 4 then it leads to the ...
- 181
4
votes
1
answer
189
views
What is a geometric demonstration of the harmonic and geometric means in Archimedes' approximation of π?
Going through Archimedes' approximation to π using the method of inscribed and circumscribed polygons, one comes upon his recursive formulae for their perimeters. Representing the perimeter of the ...
- 143
-2
votes
2
answers
97
views
$AB = AC$, $BC = 2$ and $∠BAC = 90$°. Then find the value of $\dfrac{\text{$a$}}{\text{$b$}}$. [closed]
In the figure, $AB = AC$, $BC = 2$ and $∠BAC = 90$°.
$BD = CD = 1$. $DE$ and $DF$ are the arcs whose centers are at $B$ and $C$ respectively. The area of the shaded region is $x$. If $x = a - b\pi$ ...
0
votes
1
answer
238
views
Is there any way to circumference of a circle with radius but without pi? [closed]
Please let me know if anybody knows how to calculate the circumference of a circle with radius but without pi?
6
votes
0
answers
237
views
Why does the equation of the circumference of a circle in spherical and hyperbolic space satisfy $C''=-KC$?
In a space of constant curvature $K$, the function for $C(r)$ where $C$ is the circumference of a circle of radius $r$ satisfies:
$C''=−KC$, with initial conditions $C(0)=0$ and $C'(0)=2\pi$.
(Units ...
- 24.7k
0
votes
0
answers
142
views
Circle from $n$-gon Area${}=1$
To gain more intuitive nature about the irrational and transcendental behavior of $\pi$ circles were constructed with $n$-gons circumference $C=1$ see [SE].
This method however describes how a circle (...
- 708
3
votes
1
answer
253
views
Circle from $n$-gon circumference${}=1$
To get a better intuition why: $\pi$ is irrational and transcendental I came up with the following analyses. I am a amateur but I give it my best to describe:
Imagine a piece of very flexible thin ...
- 708
3
votes
0
answers
97
views
$\pi$, from Pentadecagon - infinitely expanding Balloon nested Radical
In this post, I would like to share the findings on derivation of $\pi$ with Pentadecagon inscribed in unit circle.
Here the side of each chord is $2\sin12^\circ$
(Bisecting the chord by segment which ...
- 1,027
0
votes
0
answers
69
views
Defining a Circle using its Diameter instead of its Radius
The definition of a circle is:
Set of points in a plane that are a given distance (Radius) from a
given point (Center of the Circle)
Recently I've been hearing some discussions in the usage of π ...
- 212
3
votes
0
answers
521
views
Generating $p$-norm circles
I came across this post from 2016 and am curious about something one of the answers called "opaque". Unfortunately, that member appears to be inactive.
Measuring $\pi$ with alternate ...
- 650
-1
votes
2
answers
241
views
Is the circumference of a unit circle irrational?
Let us assume that I have a unit circle, and there are no existence of errors in measurement of a quantity. Is the circumference irrational, like the diagonal of a unit square?
Edit : My apologies. ...
0
votes
2
answers
424
views
Express in base π the circumference of a circle of radius 1. [closed]
Express in base π the circumference of a circle of radius 1.
Not sure how to approach this problem. Can you help?
0
votes
1
answer
55
views
Trigonometry issue
Hi I'm sorry i'm coming to you because i'm really bad at maths.
But i'm trying to create a visualisation for my project...
My issue is that I want to make the 4 big circles grow proportionally when ...
-1
votes
1
answer
100
views
Find the function does describe the the percentage of the area that each circle overlaps
I saw this question, yesterday and it got me thinking, what function does describe the the percentage of the area that each circle overlaps.
In that diagram it is given that the distance between the ...
user794351
3
votes
2
answers
526
views
Did Archimedes squared the circle?
What i can't understand is that I'm reading book " a History of Mathematics by boyer" and it says Archimedes made possible to construct a triangle equal in area to that of a circle by help of spirals....
- 342
2
votes
3
answers
128
views
Which Areas are closer?
Each circle-and-square pair share the same center. Which areas are closer to being equal?
Choices:
$A$ and $B$
$C$ and $D$
The difference of $B$ and $A$ is equal to the ...
1
vote
2
answers
124
views
How to solve this definite integral $ \int_0 ^a \sqrt {9-x^2} \ dx $ without using trigonometric substitution
For what value(s) of $a$ is $$ \int_0 ^a \sqrt {9-x^2} \ dx = \pi \ \text{ ?} $$
I solved this question by using integral substitution, but I was told that it is better to obtain this solution by ...
user722457
0
votes
1
answer
184
views
If Pi were equal to 3, time travel possible? [closed]
In Algebra 2 back in high school, there was a math teacher that taught at Princeton who mentioned that if pi were 3, we could be talking about time travel and also travel through dimensions.
Why ...
user754076
-1
votes
2
answers
140
views
Why is '$\frac{r^2\pi}{d^2}=\frac{\pi}{4}$'? (a.k.a. the ratio between a square and a circle, $a=d$)
When we had the circle in math (a while ago) our math teacher said we should try to find a way to calculate the area of a circle. My friend and I came up with a way to do this: you had to square $d$ ...
- 21
0
votes
5
answers
259
views
Possibility of finding exact value of circle area
I am taking a Calculus course and my current theme is calculating a circle's area from scratch, and the tutor is splitting the circle in smaller circle shapes, draws them as a rectangle and putting ...
- 109
2
votes
2
answers
153
views
Geometric Approximation for Area of Circle Using Calculus
A couple of years ago, I came up with this formula:
$$\lim_{n\to\infty}\frac{180\left(\pi^2r^2\cot\left(\frac{180}{n}\right)\right)}{n\pi}=\pi r^2$$
I derived it from a geometric perspective and just ...
- 1,084
0
votes
0
answers
116
views
Calculating $\pi$ by inscribing and outscribing $2^n$-gons
I am trying to approximate $\pi$ by using inscribed and outscribed(?) polygons. I already have a formula for the circumference of the inscribed ngons:
square= √(2×4,
octagon= √(2-√(2))×8,
16gon= √(...
6
votes
2
answers
1k
views
Formula to Calculate Each Pie Angle Where the Intercepted Arc is NOT the Center Point of the Circle AND All Slices are Equal Sized
I saw this picture titled "How to Start a Fight at Thanksgiving" and it made me laugh and then it made me wondered how to cut a pie into (N) number of pieces of equal surface area, but the central ...
- 291
1
vote
0
answers
243
views
Why does the value of pi approximate to 3.14159?
I have seen this question asked countless times online, but almost every time it is misunderstood. What property of a circle causes the ratio between the circumference and 2 times the radius (pi) to ...
- 331
0
votes
1
answer
81
views
Prove that the circumference of a circle is $25\pi$ [closed]
A regular hexagon inscribed in a circle has an area of $$54*3^\frac{1}{3} \text{sq.in}$$
Prove that the circumference of a circle is $$25\pi$$
- 155
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votes
2
answers
2k
views
What is the length of the arc on the unit circle subtended by an angle of $120^\circ$? Show all work.
What is the length of the arc on the unit circle subtended by an angle of $120^\circ$? Show all work.
$\dfrac{2}{3}$
$\dfrac{1}{3}\pi$
$\dfrac{2}{3}\pi$
$\pi$
I used an equation where the central ...
- 151
0
votes
2
answers
182
views
Confusion over the word "ratio" in the definition of $\pi$
According to Wikipedia, pi is "a mathematical constant that was originally defined as the ratio of a circle's circumference to its diameter."
However, when I think of the word "ratio", something like ...
- 21.8k
1
vote
1
answer
558
views
Can The Existence Of $\pi$ Be Proved Without Formal Analysis?
I hope this question is not too long, but I have included some extra information to clarify the context of the question
and hopefully avoid the 'circular' arguments which inevitably occur on this ...
0
votes
2
answers
103
views
Can A be proved using B when B was proved using A?
Many questions have been posted on this site about the irrationality of $\pi$; I'll be referring to one such question here.
The accepted answer mentions that $\pi$ is irrational because it is the ...
- 314
1
vote
1
answer
130
views
How to find the $\pi$? [duplicate]
We know as we increase the number of sides in regular polygon, after infinite repetition it will give us a circle. So, is there any way to find a function which approaches to value of $\pi$ when we ...
- 31
1
vote
6
answers
558
views
$\pi$ out of a right triangle and a circle
I've a right triangle that is inscribed in a circle with radius $r$ the hypotunese of the triangle is equal to the diameter of the circle and the two other sides of the triangle are equal to eachother....
- 237
0
votes
1
answer
629
views
Approximating Pi using Numerical Methods
Area of circle =r^2*pi, this circle has radius =1, so A=pi.
Split the circle into four piece and focus on the upper right one (quadrant 1).
Now I will use trapezium method and simpsons rule to ...
- 57
0
votes
0
answers
62
views
Identification the type of curve (fractal?)
First we have a square inscribed in a circle with radius $1$. By connection vertexes of this square we have two diagonals, which divides square for $4$ rectangular triangles with (at least) one corner ...
- 1,473