The radius is the defining feature of the circle............that and the point u start from..........the center........2 and one...........1/2 or 2/1............................


Fundamentals

Name

The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase Greek letter π, sometimes spelled out as pi, and derived from the first letter of the Greek word perimetros, meaning circumference.^[6] In English, π is pronounced as "pie" (/paɪ/, paɪ).^[7] In mathematical use, the lowercase letter π (or π in sans-serif font) is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation.
The choice of the symbol π is discussed in the section Adoption of the symbol π.

Definition

The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.

π is commonly defined as the ratio of a circle's circumference C to its diameter d:^[8]

${\displaystyle \pi ={\frac {C}{d}}}$

The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curved (non-Euclidean) geometry, these new circles will no longer satisfy the formula π = C/d.^[8] Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined independently of geometry using limits, a concept in calculus.^[9] For example, one may compute directly the arc length of the top half of the unit circle given in Cartesian coordinates by x² + y² = 1, as the integral:^[10]

${\displaystyle \pi =\int _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}.}$

An integral such as this was adopted as the definition of π by Karl Weierstrass, who defined it directly as an integral in 1841.^[11]