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?
$\endgroup$ 73 Answers
$\begingroup$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
$$ A_2=A_1-L\cdot w $$$$= 101.5-\dfrac{100\times 2.8\times 5280}{43560}=67.56 $$
$\endgroup$ 3 $\begingroup$If the shoreline can be taken to be a "rounded" line as in the example you have given
then you can conveniently approximate it by arcs of circle of different radii.
To this purpose you can proceed to break the shoreline into pieces with quite different curvature, by drawing the normals
at the separation like I started to do in the sketch.
Take note of the center given by the pairwise crossing points, of the relevant radius and angle, and also of the position
of the center wrt to the direction of the outer normal: for $C_1$ the in-movement translates into a positive $\Delta R$, negative instead for $C_2$.
Check if the sum of $R_k \cdot \alpha _k$ approximate well enough the total length of the shoreline or adjust the partitioning.
After which apply to each $R_k$ a $+/- \Delta R$ which is the same in absolute value and the sign is as said above.
Then you can calculate easily the area of the blue region.
The above provided that the circular sector does not close up, but if that happens you can see that at the beginning by having the relevant center in the blue area.
From here so the credit for the answer can go to the right person: Post with the original answer
The circumference of the lake is 2.8 miles, or 14684 feet. If the lake is rectangular, then going in by 100 feet on all sides reduces the area by (14684 feet - 400 ft) * 100 feet, which is 14,284,000 square feet, which is 32.8 acres. So removing this would reduce the area from 101.5 acres to 68.7 acres. The nice thing about stating the original data in terms of area and circumference is that even a slightly different shape would give similar results.
so as a formula I could plug into excel with variables C= circumference in feet, A= known surface area
A-(((C-400)*100)/43560) = remaining surface area
