Salinon Calculator

Name: Salinon - Geometry Calculator
Author: Jürgen Kummer

Calculations for a salinon, also known as Archimedes' salt cellar. The salinon is formed by a semicircle with the radius R, on whose base another smaller semicircle with radius r in opposite orientation is centrally attached. Left and right of this smaller semicircle, another semicircle with the radius s is removed so that the straight lines disappear. The salinon has the same area as the inscribed circle with the radius t and has the same perimeter as the circle with the radius R.
Enter [R and r] or [s and t] and choose the number of decimal places. Then click Calculate.

Formulas:

R = t + s

r = t - s

s = \frac{R - r}{2}

t = \frac{R + r}{2}

p = 2 \cdot π \cdot R

A = \frac{1}{4} \cdot π \cdot {(R + r)}^{2} = π \cdot t^{2}

pi:

π = 3.141592653589793 ...

Radiuses and perimeter have a one-dimensional unit (e.g. meter), the area has this unit squared (e.g. square meter).

Much like the arbelos, the salinon was presumably mathematically described and investigated by Archimedes. Both the salinon and the arbelos are formed from three circles of different sizes, two of which determine the size of the third. Their areas can be expressed in terms of a fourth circle that is found within these figures.

Unlike the arbelos, the salinon is symmetrical. It possesses an axis of symmetry passing through the center of the small circle and perpendicular to the straight edges of the large semicircle and the lateral semicircles.

The word salinon likely derives from the Ancient Greek word for a salt cellar, a shallow dish featuring a central boss, which, of course would have rested on a table in an orientation inverted from that shown in this drawing and typically depicted when illustrating this geometric form. The salinon is of particular interest from a historical perspective, for the history of mathematics and geometry. The mere fact that Archimedes devoted his attention to it lends this figure a certain significance. The salinon appears to have no known practical applications. However, similar curved forms may be encountered in design and ornamentation.

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