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circle Wikipedia and it's perimeter and area
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perimeter of the circle ::::

The perimeter of a circle, often called the circumference, is proportional to its diameter and its radius. That is to say, there exists a constant number pi, π (the Greek p for perimeter), such that if P is the circle's perimeter and D its diameter then,

P = π ⋅ D . {\displaystyle P=\pi \cdot {D}.\!} P=\pi \cdot {D}.\!

In terms of the radius r of the circle, this formula becomes,

P = 2 π ⋅ r . {\displaystyle P=2\pi \cdot r.} {\displaystyle P=2\pi \cdot r.}

To calculate a circle's perimeter, knowledge of its radius or diameter and the number π suffices. The problem is that π is not rational (it cannot be expressed as the quotient of two integers), nor is it algebraic (it is not a root of a polynomial equation with rational coefficients). So, obtaining an accurate approximation of π is important in the calculation. The computation of the digits of π is relevant to many fields, such as mathematical analysis, algorithmics and computer science.

Area of the circle ::::

In geometry, the area enclosed by a circle of radius r is πr2. Here the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.

One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons. The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and the corresponding formula–that the area is half the perimeter times the radius–namely, A = 1/2 × 2πr × r, holds in the limit for a circle.

Although often referred to as the area of a circle in informal contexts, strictly speaking the term disk refers to the interior of the circle, while circle is reserved for the boundary only, which is a curve and covers no area itself. Therefore, the area of a disk is the more precise phrase for the area enclosed by a circle.