Hans Walser, [20140111]
Regular
polygon in the square
We inscribe a regular polygon in the square such that one side of the polygon is parallel to a diagonal of the square. Figure 1 depicts the situation for n = 11.
Fig. 1: Elevengon in the square
The angle is close to 150ˇ/11 = 13.6364ˇ.
One may think that for we get the limit:
ThatŐs not right. To see this, we consider the special case of regular ngons where n is a multiple of 8 (Fig. 2 for n = 24).
Fig. 2: 24gon in the square
In this special case itŐs easy to inscribe the polygon in the square. For the angle we have in this special case:
Table 1 gives some examples.
n 


8 
19.47122064ˇ 
155.7697651ˇ 
16 
9.420172938ˇ 
150.7227670ˇ 
24 
6.242752283ˇ 
149.8260548ˇ 
32 
4.672340688ˇ 
149.5149020ˇ 
40 
3.734283802ˇ 
149.3713521ˇ 
48 
3.110281217ˇ 
149.2934984ˇ 
56 
2.665117811ˇ 
149.2465974ˇ 
64 
2.331502714ˇ 
149.2161737ˇ 
72 
2.072157264ˇ 
149.1953230ˇ 
80 
1.864755156ˇ 
149.1804125ˇ 
88 
1.695106618ˇ 
149.1693824ˇ 
96 
1.553760358ˇ 
149.1609944ˇ 
104 
1.434177567ˇ 
149.1544670ˇ 
112 
1.331690076ˇ 
149.1492885ˇ 
120 
1.242875923ˇ 
149.1451108ˇ 
128 
1.165169469ˇ 
149.1416920ˇ 
136 
1.096609254ˇ 
149.1388586ˇ 
144 
1.035670029ˇ 
149.1364842ˇ 
152 
0.9811478612ˇ 
149.1344749ˇ 
160 
0.9320797475ˇ 
149.1327596ˇ 
1000 
0.1491172888ˇ 
149.1172888ˇ 
1000000 
0.0001491168825ˇ 
149.1168825ˇ 
Tab. 1: Examples
We see, that the limit of seems not to be 150ˇ.
In fact, the limit is in our case:
We can prove this using the rule of Bernoulli – de lŐH™pital:
In our case we have .