Why does pi have a constant value of 3.14159265359...?  can we define other values for pi? 

$\pi$ is a lowercase greek letter used as a mathematical constant pronounced as "pi", It is defined as the ratio of a circle's circumference to its diameter. It is widely used in many mathematical and physics formulas and also has other equivalent relations. $\pi$ lowercase usage is quite different from the uppercase usage $\Pi$ which is used to denoted mathematical product of sequences. 
$\pi$ is also known as the archimedean constant named after the ancient mathematician Archimedes of Syracuse.
$pi$ belongs to the family of irrational numbers but "can not" be expressed as a common fraction although it has been approximated to $\frac{22}{7}$. It has an infinite decimal fraction thereby making the decimals unending and can only be approximated to certain decimal places of choice.
For more than a thousand years, different approximations and definitions of pi has been given ranging from one generation to another because of its complexity, it is a transcendental number and can not be defined as a root of any polynomial with a known rational coefficient. Ancient mathematicians have given different methods of approximation for pi, but which of these approximations is widely accepted today?
In 250 BC, Archimedes of Syracuse developed an algorithm for the approximation of $\pi$ with an arbitrary accuracy. The Chinese mathematicians made a seven digit approximation while Indian mathematicians made a five digit approximation. About a thousand year later, the first exact formula for $\pi$ was discovered using infinite series and later in the 14th century, the Madhava-Leibniz series was invented by Gottfried Leibniz and indian mathematician Madhava of Sangamagrama.
 
$\pi$ today has become more complicated due to invention of modern computations which has helped in developing algorithms for calculations of hundreds of digits of $\pi$ without stress.



The figure above shows a circle where its circumference is denoted by $C$ and its diameter $d$, the relationship between $C$ and $d$ is represented by $\pi$ which is defined as thus:
$\pi=\frac{C}{d}$ where $C=22$ and $d=7$, the ratio $\frac{C}{d}$ is constant regardless of the circle's size. Therefore, if a circle has twice the diameter of another circle, it will also have twice the circumference, preserving the ratio $\frac{C}{d}$. This concept is defined by the Euclidean geometry. If a circle can be extended to any curve, it becomes non-euclidean and therefore $\pi=\frac{C}{d}$ will fail to hold.
Approximations of $\pi$ can be defined arbitrarily as far as possible, as far as it tends to $3.14159265359...$, the commonly used values of $\pi$ are $3,\frac{22}{7},\frac{333}{106},$ and $\frac{355}{113}$ These numbers are among the best-known and most widely used historical approximations of the constant, you can generate many numbers that are approximately close to $\pi$ using any algorithms of numerical analysis, because of the irrationality of $\pi$ no common fractions can have exact values becuase it has infinite number of digits in its decimal representation and does not settle into infinitely repeating patterns of  digits.
In addition to the transcedental and irrational properties of $\pi$, it is given that it can not be written as a square root or nth root of any number and that is why squaring a circle is mathematically near impossible.
The first 50 approximation of $\pi$ is $3.14159265358979323846264338327950288419716939937510...$ it can be written in base 2 as $11.001001000011111101101010100010001000010110100011...$
$\pi$ can also be defined differently in trigonometry, complex numbers but that is not a topic of discussion for today.