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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. Hypothetical Lake Shape

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?

Alith7
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  • This is impossible, without further assumptions. You could model all lakes to be ellipses (say), and that would give you a unique and consistent answer.... but it would not be accurate. – David G. Stork Aug 13 '20 at 19:17
  • What table is required for given surface area and shoreline boundary length. Do you have several boundary coordinates available in a table? – Narasimham Aug 13 '20 at 19:32
  • @Narasimham I do not. – Alith7 Aug 13 '20 at 19:41
  • Honestly, if we are talking about a real lake, Google maps can measure areas pretty accurately. But as a math question, there are'nt enough details for an accurate solution. – Eminem Aug 13 '20 at 20:26
  • I'm sure Google can, but I don't have the time or resources to do this for even the 700ish lakes out of the 15,000 that I think will fall into the window. – Alith7 Aug 13 '20 at 21:01
  • If $100$ feet is small compared to the diameter of the lake you can approximate the area to be subtracted as a ribbon whose length is the shoreline and whose width is $100$ feet. – John Douma Aug 13 '20 at 21:07
  • I used the same idea only figuring sides of a rectangle with the same perimeter and area, subtracting and calculating the new area, and that seems to match the original solution I posted for the lakes that I spot checked. Obviously figuring it out for an oval would be more accurate, but not having that formula, this will have to do. – Alith7 Aug 14 '20 at 21:31

3 Answers3

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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

$$ A_2=A_1-L\cdot w $$ $$= 101.5-\dfrac{100\times 2.8\times 5280}{43560}=67.56 $$

Narasimham
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If the shoreline can be taken to be a "rounded" line as in the example you have given

Lake_shrink_1

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.

G Cab
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  • This looks to be the most accurate answer, but also not useful for my purposes. I only need rough approximations of thousands of lakes. This just isn't feasible for the amount of data I need to process. – Alith7 Aug 14 '20 at 13:28
  • @Alith7: then you need a statistical approach . Take a representative sample, fix some parameters (width, height or some more chords), and the variable of interest. Try and fix a function representing the variable. – G Cab Aug 14 '20 at 16:29
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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
Alith7
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  • This answer is invalid if the lake isn't rectangular (and no natural lakes ever are). You choose a model shape: rectangular, oval, ... and you can answer your question consistently, but not accurately. You state explicitly the lake has "irregular shape," so a rectangle is the first shape to exclude. – David G. Stork Aug 13 '20 at 19:34
  • I understand that it is invalid if I were needing exact data, but I need rough data for comparison purposes. Yes, to be accurate, the exact shape of each lake would have to be entered, the 100ft off shore would have to be calculated and mapped, and then the new are figured out. Since I KNOW the current area with a given shoreline length, this method of rough approximation will work for my purposes. – Alith7 Aug 13 '20 at 19:45
  • basically, I have a database of 15,000 lakes, I need to illustrate that if this rule is enacted approximately X number of these 15,000 lakes will be affected. No one is going to nitpick that some of the numbers aren't exactly right. – Alith7 Aug 13 '20 at 19:46
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    Then I would fit the data with ovals, where the major and minor axes determine the area and perimeter length uniquely. – David G. Stork Aug 13 '20 at 20:03
  • that actually gives me a thought! I know the area and the perimeter, if I could figure out a formula to take that info and get the rough 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. And at this point, by math skills are failing though. – Alith7 Aug 14 '20 at 16:41
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    No need to get a "rough" major and minor axis, you can calculate the exact values. (Again, all this is under the fairly reasonable assumption of elliptical lakes.) – David G. Stork Aug 14 '20 at 16:42