I have to put some data together for a presentation on lakes that will be affected by a rule change. I have a table with the surface area of the lake and the length of shoreline. I need to calculate the inner surface area of the lake if I move in 100ft from the shore line. Obviously lakes are irregular in shape, so I'm not even sure if this is possible.
Essentially, lets say the circumference of the lake has 2.8 miles of shoreline and has 101.5 acres of surface area. How would I calculate the surface area left if I measure 100ft in from the shoreline?
See graphic. The blue shape is the hypothetical 2.8 miles of shoreline with 101.5 acres of surface area. I need to come in 100ft all the way around the lake, and figure out what the remaining surface area is illustrated roughly by the green shape. *** This is just a random drawn shape, and is not meant to be measured.
EDIT
Courtesy of David G. Stork in the comments below, I think I know what I need, just need help with the formulas.
Since I know the area and the perimeter, if I could figure out a formula to take that info and get the major and minor axis for an oval, I could take that answer, plug it into another formula that would subtract the 200 feet off each axis, and then recalculate the remaining area. Should get me close enough for comparison purposes. And at this point, by math skills are failing though. Is there someone MUCH smarter than I that could help with these two formulas please?
Best Answer
The rate at which an area grows/shrinks equals boundary length times constant boundary width. This is accurate enough for differentials of convex boundary shapes,i.e., if $w<<L$.
When reduction is $34$% its accuracy is poor as in this case.
Area remaining in acres
$$ A_2=A_1-L\cdot w $$ $$= 101.5-\dfrac{100\times 2.8\times 5280}{43560}=67.56 $$