Approximate the Circle Constant

This example approximates the value of the circle constant (tau) using a formula to approximate the circumference of a circle.

Steps

  1. Start with the definition of the circle constant , where the constant is defined as the length of the circumference of any circle divided by the length of its radius.

    Tau Definition

    This gives the expression below as a starting point.

  2. Observe that the circumference of the circle can be approximated as the perimeter of a regular polygon denoted where is the perimeter and is the number of sides. As the number of sides increases so does the accuracy of the approximation.

    Tau Approximation
  3. At this point, it is fair to wonder if we have swapped one hard problem for another equally hard problem. Luckily, there is a formula for a special case of regular polygons with sides , , , etc. The perimeter of these regular polygon is written as to represent that the number of sides corresponds to powers of .

    Set and substitute the approximation into our expression from step one.

    The first couple regular polygon approximations are shown below. The correct digits of are highlighted green.

    Polygon Formula Approximation
    2^2 sided polygon
    2^3 sided polygon
    2^4 sided polygon
    2^5 sided polygon
    2^6 sided polygon

    While this approximation doesn’t converge very fast, as gets increasingly big, the approximation does approach the value of the circle constant .