Get hold of any good land surveying textbook.
Look up some sample hand calculations for balancing a boundary survey and using the x,y (northings and eastings as Surveyors would say) to calculate the enclosed area.
If circular curves are involved you may need to do some side calculations.
Then, write your own spreadsheet based on an example to which you already know the answer. This will allow you to check your spreadsheet to make sure it is working correctly.
Once it is, you can use it to calculate any area for which the boundary can be defined.
That's how they taught me to do it when I was in school 43 years ago, but I don't think geometry has changed in several centuries now.
good luck