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Leonardo da Vinci (1452-1519) was many things: painter, physicist, engineer, anatomist... and amateur mathematician. He was not a methodic writer, but now and then he would note down a construction procedure for some regular polygon. (Some of them he later considered bad enough to label them “falso”.)

We will review here his construction of the regular pentagon . (It is approximate; an exact one was given by Euclid.)

Let the given side of the regular pentagon be *DA=a*. Drawing circles of radius *a*, centered at *D* and at *A*, we obtain points *P* and *R* on the perpendicular bisector of *DA. * We then divide *DA* into eight equal parts. We now draw *PG* ||*AD*, with *PG = AD*/8; *AG* and *PS* intersect at *O*, center of the circle circumscribed about the sought pentagon. From here, we can proceed easily: it is only necessary to copy angle *AOD* = 2p/5 four more times.

How accurate is this procedure? To find out, we need only note that since triangles *SAO* and *PGO* are similar, it follows that *AS*/*PG = SO*/*OP =* 4/1. And since triangle *DAP* is equilateral of side *a*, *SP*, the altitude of this triangle, has length *a* √3/2. Then, the tangent of angle *SOA* is equal to the tangent of angle *GOP. * This tangent is easily calculated to be

*PG*/*OP =* ( *a*/8)/(a√3/10) = (5/12)√3.

Therefore, the sine of angle *SOA* is (5/73)√73 = 0.585.

This is, in fact, a good approximation, as it makes sin 36° = 0.585 instead of 0.587. [1] Of course, 36° = (1/2)(360°/5).

Raul A. Simon, "Approximate Construction of Regular Polygons: Two Renaissance Artists - Leonardo da Vinci," *Loci* (August 2010)