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Archimedes developed exact methods for geometric measurement. In Measurement of a Circle by Archimedes, he proves the relation between a circle's area, radius and circumference and obtains rational bounds on its circumference-to-diameter ratio.^[ref]

The bounds $223/71<\pi <22/7$ come from comparing inscribed and circumscribed polygons. Increasing the number of sides traps the circular quantity between computable geometric quantities. The argument illustrates the power of a rigorous bound: one need not express a number exactly to establish an accurate approximation with a guaranteed error.^[ref]

The Rayleigh distribution is the probability distribution of the Euclidean norm of a two-dimensional random vector with independent zero-mean normal random variables of equal variance as components. If $X,Y\sim N(0,{\sigma }^2)$ are independent, thenHere $\sigma$ is the standard deviation of each component, not of the radius $R$. At $\sigma =1$ this gives the unit-scale probability density function $r{e}^{-r^2/2}$.^[ref]

This geometry turns two signed errors into one nonnegative distance. For example, suppose horizontal and vertical positioning errors are independent $N(0,4{\mathrm{cm}}^2)$ random variables. Their means both vanish, but the distance from the intended point is usually positive. The Rayleigh distribution with $\sigma =2\mathrm{cm}$ gives the probability of landing inside any centered disk. These are chosen model assumptions, rather than measurements of a particular instrument.

An apparently paradoxical feature is that the most likely location per unit area is the origin, while the radial mode is $\sigma$. A narrow annulus farther out has more area. Near zero that growing circumference outweighs the falling planar density; farther out the exponential function wins. Distinguishing densities measured against different elements of area or length resolves the paradox.