On an infinite 1 by 1 grid, we want to make an enclosure with 20 fences that are each 5 units long. The two ends of each fence has to be on a node of the grid. What is the maximal area of the enclosure? (Fences can cross nodes and the enclosure has to be a polygon.)
Largest area: 768
The largest (I've found) is as follows:
I believe this will be the maximum, if not very close to it, as the theoretical maximum area is only $789.22$ (see further down for why).
Here are some other attempts:
Starting simply, the area of a square with side length of $25$ is $625$:
Next is my favourite, a snowman. Sadly he has quite a small area, but I'll add him here anyway. This shows that adding lots of small polygons doesn't work.
And a diamond is big , at $736$:
And I initially overlooked an even larger one at $741$:
But the largest will be when the shape is as circular as possible (see further down for why), which is this shape at $768$:
I'm using this website to draw these, which very helpfully tells you the length of the line you are drawing.
Little bit of maths, that got me thinking how to solve this:
From a node, there's only a few other nodes that can be connected such that the length is $5$ units. Using Pythagoras's theorem we can see only a couple of these 'possibilities' have a length of $5$:
So from one node (in the first quadrant), these are the possible lines that can be drawn:
The greatest possible area, therefore, if we didn't have to connect to nodes would be $789.22$
The internal angle of an icosagon is $(180 \times (n-2)) / n = 3240/20 = 162$ degrees. So the maximum area will be when these fences are set so they have an angle of $162$ degrees between each one. However this wont be possible.
Going back to the possibilities, and using some trigonometry, the angles are:
As you can see, it won't add up to $162$, no matter how you place them. There are also no n-sided shapes with internal angles of $126.87$ or $143.13$ either.
So to get the biggest area, the largest possible n-sided polygon needs to be formed. This means the largest area will be when the shape is as round as possible.
This is the basis which I am using/used to find large areas.
Best I've done is 757 which looks like this:
My process was to start with a 5 fence by 5 fence square and then push out the middle bits by using 3x4 diagonals to make the polygon more circle shaped. If you push out the middle bits another time you get 753 which is close, but slightly worse.