The Gemara states that Avraham davened Minchah “when the [eastern] faces of the walls begin to darken,” i.e., immediately after solar noon. Now, we know that our prayers correspond to the Tamid-offerings, and that the time period allowed for the Minchah prayer is identical to that allotted for the afternoon Tamid. It should emerge that the afternoon Tamid may be brought “when the [eastern] faces of the walls begin to darken” yet we find that the earliest permissible time for the afternoon Tamid is not until half an hour past noon! There is no contradiction because, unlike standard walls, the eastern faces of the Beis Hamikdash walls only fell into shadow at half an hour past noon. Rashi here explains that this resulted from the Beis Hamikdash walls being thicker at their base than at their top; since they tapered as they rose, the sun continued to shine on their eastern faces even past noon, and only at half an hour past noon did the eastern faces finally darken in shadow.
Even though the position of the sun changes throughout the year, the only time that the Tamid-offering was permitted to be brought as early as half past noon is when erev Pesach is on a Friday (to allow enough time for the multitude of Pesach-offerings to be brought and roasted before Shabbos). As a result, the Beis Hamikdash walls were designed to darken at half past noon specifically on the fourteenth of Nisan (the approximate date of the Spring equinox).
The position of the sun at half an hour past noon on the Spring equinox in Yerushalayim is a readily quantifiable phenomenon, making it possible — using the diagram below — to estimate the slope of the walls of the Beis Hamikdash.