Is Pi, the mathematical structure of a circle? ?

no, it's just an irrational number, one of an infinite number of such.

π =3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209  74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651  32823 06647 09384 46095 50582 23172 53594 08128 48111 74502 84102

NO!!! 

pi is the constant of proportion of a circle's diameter to its circumference 

For all circles however large or small, the circumference is directly proportional to the diameter. 

C is proportional to d 

We then insert  'k' the constant of proportion

C = kd 

The value of 'k' is found to be 'pi = 3.14159 26.... 

So the equation can be written as 

C = 3.1415926....d

or  

C = pi*d 

NB 'pi' is an irrational number, which casually means that the decimals go to infinity and the digits are are not in any regular order.  For everyday school work pi = 3.14 or 3.1416. However these are only approximations. 

NNB 'pi' is always spelled in small case letters, NOT 'Pi' , but 'pi' . It is the small case Classical Greek letter 'p' and means 'proportion;. 

NNB As an aside do not spell 'pi' as 'Pie / pie' , that is the foodstuff that you eat as in meat pie , apple pie. 

In part, yes.  It involves the integral of a rectangle with sides r and C/2.

The number π (/paɪ/) is a mathematical constant. It is defined as the ratio of a circle's circumference to its diameter, and it also has various equivalent definitions. It appears in many formulas in all areas of mathematics and physics. It is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, and is spelled out as "pi". It is also referred to as Archimedes' constant.

It's a number.  But it so happens that the distance around a circle's edge (the "circumference") is the number pi multiplied by the length of the circle's diameter.  It makes sense that there is just one such number -- because all perfect circles have exactly the same shape.

It's a measurement.