Articles of pi

Relationship between the sides of inscribed polygons

In my math textbook there’s a demonstration for the calculus of the circumference of a circle that involves regular polygons inscribed in the circle, but I don’t get it. The book gives the following formula: $$l_{2n}=\sqrt{2R^2 – R\sqrt{4R^2-l_{n}^2}}$$ Where $l_{n}$ is the side of a regular n-sided polygon inscribed in a circle with R radius. […]

Calculating custom bits of PI in hex or binary without calculating previous bits

I tried some spigot formulas to calculate custom hexadecimal PI digits. But any formula I tried definitely needed iterating and calculating sum from i=0 to N to get N-th digit. How to get N-th hex digit without calculating previous digits?

Riemann sum formulas for $\text{acos}(x)$, $\text{asin}(x)$ and $\text{atan}(x)$

In this post just another $\pi$ formula, I gave a kind of Riemann sum to compute the area of a quarter of circle based on a very simple geometric trick, and same reasoning can be used to compute any inverse trigonometric formulas based on the angle area. But I think that because of the simplicity […]

Finding sine of an angle in degrees without $\pi$

The following is found using a combination of: (a) a polygon with an infinite number of sides is a circle, (b) the perimeter of that polygon is the circumference of the circle that it becomes (of course), (c) the sine theorem, and (d) the ratio between the circumference of a circle and the diameter is […]

All $11$ Other Forms for the Chudnovsky Algorithm

Continued from this post Ramanujan found this handy formula for $\pi$$$\frac 1\pi=\frac {\sqrt8}{99^2}\sum_{k=0}^{\infty}\binom{2k}k\binom{2k}k\binom{4k}{2k}\frac {26390k+1103}{396^{4k}}\tag1$$ Which is related to Heegner numbers. Sometime after, the Chudnovsky brothers came up with another $\pi$ formula$$\frac 1\pi=\frac{12}{(640320)^{3/2}}\sum_{k=0}^\infty (-1)^k\frac {(6k)!}{(k!)^3(3k)!}\frac {545140134k+13591409}{640320^{3k}}\tag2$$ And according to Tito, $(2)$ has a total of $11$ other forms with integer denominators. Question: What are all $11$ […]

Is $\pi$ approximately algebraic?

As we know, $\pi$ is transcendental, meaning that there is no rational numbers $a_0,\ldots,a_n\in\mathbb{Q}$ such that $$a_0+a_1\pi+\cdots+a_n\pi^n=0.$$ But I was wondering if we can get this as a limiting process: Is there a sequence of polynomials $\{p_n(x)\}_{n=1}^\infty$ with rational coefficients such that the first positive root of $p_n(x)$ tends to $\pi$ as $n\to\infty$?

a Circle perimeter as expression of $\pi$ Conflict?

I know that the the perimeter of a circle is $$2\pi r$$ The problem is that $\pi$ is un-finite number. ( its decimal representation never ends) Im having trouble to understand : If I “cut” the circle and make it as a line : – and i look at this line : the line has […]

Is the value of $\pi$ in 2d the same in 3d?

I am starting with my question with the note “Assume no math skills”. Given that, all down votes are welcomed. (At the expense of better understanding of course!) Given my first question: What is meant by the perimeter of a Sector Why is the value of $\pi$ not exactly $3$? why is it $3.14$………. or […]

$ \sum _{n=1}^{\infty} \frac 1 {n^2} =\frac {\pi ^2}{6} $ then $ \sum _{n=1}^{\infty} \frac 1 {(2n -1)^2} $

If $ \sum _{n=1}^{\infty} \frac 1 {n^2} =\frac {\pi ^2}{6} $ then $ \sum _{n=1}^{\infty} \frac 1 {(2n -1)^2} $ Dont know what kind of series is this. Please educate. How to do such problems?

Does $\pi$ have infinitely many prime prefixes?

This is inspired by this question. Does $\pi$ have infinitely many prime prefixes (in base $10$)? That is, is the sequence A005042 infinite? It says on the OEIS that a naïve probabilistic argument suggests that the sequence of such primes is infinite. What argument would that be? Such primes would be examples of pi-primes.