Ratio of Circumference to Diameter

One of the most fundamental truths of (Euclidean) geometry is that the ratio of the circumference to the diameter of any circle is a constant, and that constant is called pi (denoted by\(\pi \)). Let C be the (length of the) circumference of a circle, and let d be its diameter. Then, we must have:

\[\frac{C}{d} = \pi \]

This will hold regardless of how small or large the circle is. Thus, whether a circle’s radius is 1 cm or 1 km, the ratio of its circumference to its diameter will be \(\pi \) in either case. The fact that this ratio is constant (that is, \(\pi\) is constant for every circle) can be proved, though the proof requires somewhat more advanced Mathematics.

Many of you would know that the approximate decimal value of \(\pi\) is 3.14, and the (approximate) value of \(\pi\) in fraction form can be written as:

\[\pi  \approx \frac{{22}}{7}\]

It is important to understand that this is only an approximation. \(\pi \) is actually an irrational number, and so its exact value cannot be written in rational form. An even better rational approximation of \(\pi \) is:

\[\pi  \approx \frac{{355}}{{113}}\]

Again, this is only a very good approximation, and not the exact value of \(\pi \).