Salinon

---

title: "Salinon" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["piecewise-circular-curves", "archimedes"] description: "Geometric shape" topic_path: "general/piecewise-circular-curves" source: "https://en.wikipedia.org/wiki/Salinon" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Geometric shape ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/f/f8/Salinon.svg" caption="The salinon (red) and the circle (blue) have the same area."] ::

The salinon (meaning 'salt-cellar' in Greek) is a geometrical figure that consists of four semicircles. It was first introduced in the Book of Lemmas, a work attributed to Archimedes.

Construction

Let A, D, E, and B be four points on a line in the plane, in that order, with AD = EB. Let O be the bisector of segment AB (and of DE). Draw semicircles above line AB with diameters AB, AD, and EB, and another semicircle below with diameter DE. A salinon is the figure bounded by these four semicircles.{{cite journal | last = Nelsen | first = Roger B. | date = April 2002 | doi = 10.2307/3219147 | issue = 2 | journal = Mathematics Magazine | jstor = 3219147 | page = 130 | title = Proof without words: The area of a salinon | volume = 75}}

Properties

Area

Archimedes introduced the salinon in his Book of Lemmas by applying Book II, Proposition 10 of Euclid's Elements. Archimedes noted that "the area of the figure bounded by the circumferences of all the semicircles [is] equal to the area of the circle on CF as diameter." |url=http://www.cut-the-knot.org/proofs/Lemma.shtml |title=Salinon: From Archimedes' Book of Lemmas |work=Cut-the-knot |accessdate=2008-04-15 |last=Bogomolny | first = Alexander |author-link=Alexander Bogomolny

Namely, if r_1 is the radius of large enclosing semicircle, and r_2 is the radius of the small central semicircle, then the area of the salinon is: A=\frac{1}{4}\pi\left(r_1+r_2\right)^2.

Arbelos

Should points D and E converge with O, it would form an arbelos, another one of Archimedes' creations, with symmetry along the y-axis.

References

References

1. Heath, T. L.. (1897). "On the Salinon of Archimedes". The Journal of Philology.
2. "Salinon".

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::