I've found two cities that are located almost precisely at the equator (and I know that since on equinoxes the Sun is directly overhead in those cities)- São Tomé and Libreville.
Using differences in sunrise/solar noon/sunset times (circa 11,5 minutes) I was able to determine the size of the ∠SãoTomé-NorthPole-Libreville (equal to the difference of longitudes) to be circa 2,875°
The distance between São Tomé and Libreville is 302,1 km (as the crow flies).
All of this data allows me to estimate the circumference of the equator: I can simply multiply 302,1 by 360°/2,875° and get 37 828 km.
(Yes, I'm well aware that this is a simplification. That's why I've also calculated this the correct but a bit more complicated way. I've got 37 832 km.)
Does the community of the FES agree? Should I add a more detailed description of my calculations?
EDIT: I've assumed the Earth to be flat and unipolar and the path of the Sun on equinoxes to be circular.
EDIT2: Since the cities are a little bit to north of the actual equator the definite circumference is even larger (not by much though).
This reminds me of one more measurement, which gave value very close to yours.
It will confirm you method.We can use ground speed of subsolar point (of "Lahaina Noon") for equinox to measure circumference of Equator.
Every model and every observation, regardless of the shape, agrees in one thing: Sun travels along Equator for equinox.
That is why Equator was established in the first place.
For ground distances we will use values from public records, used in reality for many purposes, commercial / industrial, military, infrastructures, transportation...
Anybody is entitled to provide any other records tested by verifiable measurements or reasonably long general public usage / scrutiny.
Let’s go to Africa, to Equator.
There are Jilib in Somalia, Kampala in Uganda, Mbandaka in Congo, and Libreville in Gabon.
Jilib: longitude – 42.78 degrees east
Kampala: longitude – 32.57 degrees east, distance from Jilib – 1136.5 km
Mbandaka: longitude – 18.29 degrees east, distance from Kampala – 1590 km
Libreville: longitude – 9.47 degrees east, distance from Mbandaka – 981.3 km
From Jilib to Libreville is 3708 km and 33.31 degrees of longitude.
It makes speed of solar noon to be 3708 / 33.31 * 15 = 1669.768 km/h.
That speed multiplied by 24 h gives circumference of Equator to be 40074.45 km
Ofcourse there is error.
Every measurement has errors.
But how big is our error?
Checking Equator circumference at shorter distances inbetween gives
Between Jilib and Kampala we calculate 1669.686 km/h, which gives 40072.48 km.
Between Kampala and Mbandaka we get 1670.168 km/h, giving 40084.03 km.
Between Mbandaka and Libreville we get 1668.877 km/h, which gives 40053.06 km.
All values are very close to each other. Our error is below 0.05% (below 500 ppm – parts per million).
Ok, we now have Equator circumference of 40074.5 kilometers.
EDIT: From Neves (Sao Tome) to Libreville (Gabon) is 325.06 km.
Longitudinal difference is 9.46°E - 6.54°E = 2.92 degrees.
Using your method, the circumference of Equator would be 325.06 * 360 / 2.92 = 40075.89 km.
Neves is at 0.36 degrees north, Libreville is 0.42 degrees north, so angle to Equator will be 1.177 degrees.
More accurate circumference would then be 40075.89 * cos(1.177) = 40067.4 km.
Both values would be off for a tad, based on accurate value of the distance between the two places.
For example, if the distance is 325.12 km (60 meters greater), equatorial circumference will be (325.12 * 360 / 2.92) * cos(1.177) = 40074.8 km.
Those 60 meters could be from one to another side of the central square in any or both of the towns.
Or "from dome to dome" instead of "door to door" of their respective city halls.
Longitudes and longitudinal difference have more decimals and it also affects accuracy...
But all those values are within ±0.1% and the "exact value" isn't far from them.