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

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.

This gives the expression below as a starting point.

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.

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 this 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 of approximations are shown below substituting larger and larger values in for . The correct digits of are highlighted green.

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