Hi Max,
So, as usual with this kind of stuff, it's more complicated that what it appears to be at first glance.
We do store a value for surface, but that is derived from a geometry that is not "official" per say, it's an estimated area, not EXACTLY the one that could be (or have been) measured by surveyors.
With that said, the data owner doesn't want to expose this (yet), as it's an unofficial value that lacks accuracy and the "official" stamp.
On the other hand, we have tried a couple parcel on our side, and calculated the area and compared it to the official one, if you are no surveyor (with this knack for precision to the absolute limit) you should be able to live with it.
Regarding calculating the area of the polygon, if you are developing your tool/app with Javascript, there is already plethora of well established (and mathematically accurate) library to help you with geometry manipulation, one such example would be
https://turfjs.org/
Best regards,
Pascal