Circumference and Area of a Circle

If one wants to calculate the circumference or the area of a circle with the methods of elementary geometry, then one must look for polygons which differ as little as possible from the given circle. A well-known method is to start with one inscribed and one circumscribed regular quadrilateral or hexagon, and then keep doubling the number of vertices. This gives increasingly accurate estimates of the circumference and area of the circle.

This HTML5 app performs the necessary calculations with a precision of 50 decimals. It refers to the unit circle; the radius is therefore assumed to be r = 1. The number of vertices used for the polygons can be controlled by mouse clicks on the buttons. With the three radio buttons you can select whether the side length, the circumference or the area of the polygons should be displayed.

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For a circle with radius r the following formulas result:

Circumference:C = 2 r π
Area:A = r2 π

The number π appears in both formulas. The first 100 decimal digits are given here:

π = 3.1415926535 8979323846 2643383279 5028841971 6939937510
          5820974944 5923078164 0628620899 8628034825 3421170679 ...