At solar noon the sun stands 90° minus your latitude plus the sun’s declination above the horizon: add 23° in June, nothing at the equinoxes, subtract 23° in December.
The sun is overhead at the latitude equal to its declination, and every degree you are away from that lowers it by a degree. In London (51.5°N) it is 90 − 51.5 + 23.4 = 62° at the June solstice, 90 − 51.5 = 38.5° at the equinoxes and 90 − 51.5 − 23.4 = 15° in December. A post casts a shadow of 1/tan(height): 0.5 times its height in June and 3.7 times in December.
It holds only at solar noon, which is rarely 12:00 on the clock. In the southern hemisphere the sun is to the north, and the same formula gives its height with the signs reversed. Close to the tropics the sun can pass the zenith and the formula, with the absolute value, still gives the height but the shadow flips side. The declination moves by up to 0.4° a day around the equinoxes, so use the actual date.
Estimate with the rule, then check it against the calculator that models it properly.
Open Sun Phase →