# How do we know the ratio between circumference and diameter is the same for all circles?

The number $\pi$ is defined as the ratio between the circumeference and diameter of a circle. How do we know the value $\pi$ is correct for every circle? How do we truly know the value is the same for every circle?

How do we know that $\pi = {C\over d}$ for any circle? Is there a proof that states the following: Given any circle we know that $\pi = {C\over d}$. Doesn’t such a statement require a proof considering $\pi$ is used so widely on problems involved with circles, spheres, etc. How do we truly know that the value $\pi$ is correct for all circles?

This is not a very rigorous proof, but it is how I was taught the fact that the circumference of a circle is proportional to its radius. Consider two concentric circles as in the diagram above. The radius of the smaller one is $r$, while that of the larger one, $R$; their circumferences are $c$ and $C$ respectively.

We draw two lines through the center to meet each circle, forming two triangles as shown. The ratio of their sides $r/R = r/R$, and they have a common angle $\alpha$, so they are similar. Thus $k/K = r/R$. Also note that if $\beta$ denotes the full (360 degree) angle of a circle, then $\beta/\alpha \cdot k \approx c$ and $\beta/\alpha \cdot K \approx C$.

We can say that $\frac{c}{C} \approx \frac{\beta/\alpha \cdot k}{\beta/\alpha \cdot K} = \frac{r}{R}$. As the angle $\alpha$ becomes smaller and smaller (tending towards zero, to make a limiting argument) the approximations $\beta/\alpha \cdot k \approx c$ and $\beta/\alpha \cdot K \approx C$ grow more accurate. In the limiting case — and this is where the ‘proof’ is slightly nonrigorous — we get that $\frac{c}{C} = \frac{r}{R}$.

Thus $c/r = C/R$ or equivalently $c/(2r) = C/(2R)$: the circumference divided by the diameter is always a constant for any two circles since any two circles can be made concentric by a trivial translation. We call this magic constant $\pi$.