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: Eleven-gon 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 n-gons where n is a multiple of 8 (Fig. 2 for n = 24).

Fig. 2: 24-gon 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 .