What is the real curvature of a circle?

Question

Wiki says that the curvature of a circle is 1/r, but if we consider the radius in meters or centimeters or other we get different results.

How do we know the right value?

Answer 1

The curvature $\kappa$ is "tangent direction change per length increment", and has dimension ${1\over{\rm length}}$. It follows that for a physical curve in our ambient space the numerical value of $\kappa$ depends on the chosen length unit.

When you have a circle drawn on a paper with length unit marked in the figure (say, on the coordinate axes) then it is agreed that the unit circle has $\kappa=1$ for all students in the class, independently of the sizes of their figures.

Note that you have the analogous problem already with lengths. It has nothing to do with the sophisticated notion of curvature.

Answer 2

You'll get the right answer no matter which units you use.

Suppose you have a circle whose radius is 100 inches, which is the same as 254 centimeters or 2.54 meters. Then the curvature of the circle is

$$\frac{1}{100\ \mathrm{inches}} = \frac{1}{100}\ \mathrm{inches}^{-1},$$

but the curvature of the circle is also

$$\frac{1}{254\ \mathrm{cm}} = \frac{1}{254}\ \mathrm{cm}^{-1},$$

and the curvature of the circle is also

$$\frac{1}{2.54\ \mathrm{m}} = \frac{50}{127}\ \mathrm{m}^{-1}.$$

All of these answers are correct, because $\frac{1}{100}\ \mathrm{inches}^{-1}$, $\frac{1}{254}\ \mathrm{cm}^{-1}$, and $\frac{50}{127}\ \mathrm{m}^{-1}$ are all the same quantity.