Astro Torah: Needing a GPS to Build a Sukkah

See R' Ari Storch's Needing a GPS to Build a Sukkah

Let us examine the mathematics behind this. We will find that they are slightly troubling.

As he states, the גמרא asserts that according ר' זירא the 20 אמות limit would only apply to a small Sukkah of 4x4 אמות. I know I have heard but can't remember where, that the specific issue that ר' זירא had was that the סכך needs to provide shade on the ground. If the walls of a small sukkah were 20 אמות high, there would never be shade on the ground provided by the סכך. This is certainly true. But according to the trigonometry of it, the limit should be far less than 20!

As stated by R' Ari, the highest angle the sun will reach in ארץ ישראל during this time is approximately 58°. The tan of 58° is approximately 1.6. That is to say that at a 58° angle, the opposite side (the wall) is about 1.6x the length of the adjacent (the ground.) That said, in a 4x4 sukkah, a wall of merely 6½ אמות would cause the walls to constantly cast a complete shadow on the ground and the סכך will never provide this shade. If we approach it from the other end, 20 אמות walls would prevent סכך-shade in any sukkah up to 12½ (20/1.6) אמות.