I had never heard of a planimeter[1], but what a fascinating device it is! I'm surprised that such a thing is possible: that you can calculate the area of an arbitrarily complex shape just by tracing the perimeter with a simple mechanical device. Unfortunately, how it works is not at all intuitive but seems to require a fair amount of math to understand.
This kind of thing is one of the more important ideas from 19th century math/physics, and is studied by most STEM undergraduates in their first or second year.
Not only, it was (is) extremely accurate if used properly.
I am old enough (unfortunately) to have used planimeters professionally in the '80's (just before drawing started becoming digital), at the time they were largely used because they were much faster than calculating exactly (and already hand calculators with trigonometric functions existed) some data in roads cross-sections.
So, for temporary accounting, planimeter based measures of areas were accepted (usually reduced by 5% or 10%).
Basically you had a (hand drawn) inked drawing of each cross section of the road (usually one every 20 m or so) where you would draw a pencil line at the level that was reached in the works, and then use the planimeter to get the area (and consequently volume) of the earthworks done at the date.
When the level was final, the amount would be calculated analitically, still manually, usually, but I managed somehow to write for the HP97[1][2] (constrained by 256 steps!) a small program capable of automatically calculating that, of course typing the measures manually.
Large stretches of the road earthworks had already been calculated by the planimeter and when re-calculated, the differences were very small, within 1% or 2%.
[1] http://www.hpmuseum.org/hp6797.htm [2] http://www.hpmuseum.org/97.jpg
https://en.wikipedia.org/wiki/General_purpose_analog_compute...