You calculate the interior angle in degrees using the formula ((N-2) x 180)/N.
For a four sided regular polygon (spoiler alert: a square) has an interior angle in degrees of ((4-2) x 180)/4 = 90. So you draw a four sided regular polygon by going a certain distance, turning right to make a corner with a 90 deg interior angle and going the same distance again, repeat until you get back to the starting point X:
That's easy enough, you can do the same for three sides, or five or six or anything until you approach a circle.
But this is maths and N can be any number you like. So let's try two and a half sides.
Interior angle in degrees = ((2.5 - 2) x 180)/2.5 = 36
Let's repeat the process, draw a line, turn right so the interior angle is 36 degrees and draw another line, then keep going until you get back to your starting point:
That is not a 'pentagram', it is a regular polygon with two and a half sides.
It is important to note that the edges do not actually touch each other where they appear to cross, those are not real corners, it just seems that way if you reduce this shape to a two-dimensional representation.
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There is a formula for calculating the number of diagonals, which is (N x (N-3))/2.
So a square has (4 x (4-3))/2 = 2 diagonals.
And a two and a half sided one should have (2.5 x (2.5-3))/2 = negative 0.625 diagonals i.e. none.
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There is a formula for calculating the surface area (from here). The area of any regular polygon is given by the formula: Area = (a x p)/2, where a is the length of the apothem (the line joining the half way point of a side and the centre) and p is the perimeter of the polygon.
Let's assume our square has side length 10, the perimeter is 40 and the apothem is 5. Area = (5 x 40)/2 = 100.
Calculating the length of the apothem of a two and a half sided polygon is a bit trickier, so I won't bore you with the workings, but if the side length is 10, then the apothem is 1.62.
The surface area is therefore (1.62 x 25)/2 = 20.3. The surface area of a three-sided regular polygon with side length 10 (=equilateral triangle) is 43, and the surface area of a two sided polygon (a straight line) is zero, so 20.3 looks about right to me.
This is a different answer to calculating the surface area of a five pointed star, which you can calculate the area the hard way by breaking it down into five triangles and a pentagon.
Area of each triangle = base x height/2 = 2.4 x 3.6/2 = 4.32 x 5 = 21.6
Area of pentagon in the middle = apothem x perimeter/2 = 1.62 x 12/2 = 9.7.
Total area of a similarly shaped five pointed star = 31.3.
(If I had to take a wild guess, I would say that the difference (i.e. 31.3 - 20.3 = 11) is because what appears to be the pentagon in the middle of the two and a half sided polygon does not actually exist, it has no real corners, but something went wrong in my workings, which shows area of pentagon = 9.7)
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You can also have a regular polygon with three-quarters of a side.
Interior angle in degrees = ((0.75 - 2) x 180)/0.75 = 300. What you end up with looks like an equilateral triangle, but the area of the polygon is everything outside the triangle, not inside it.
(You get exactly the same result if you try N = -3, an equilateral triangle in the middle and everything outside it is the polygon.)
Sadly, this doesn't work very well with other fractions, if N is 1/3 or 1/2, the interior angle is a multiple of 180 degrees, so it is just a straight line going there and back.
If N is 2/3, the interior angle is 360 degrees, so it's just a straight line going on forever and you never get back to the starting point.
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