99. Figs. 36, 37 show how a regular decagon, and a regular dodecagon, may be obtained from a pentagon and hexagon respectively.

Fig. 36.

The main part of the process is to obtain the angles at the center.

In Fig. 36, the radius of the inscribed circle of the pentagon is taken for the radius of the circumscribed circle of the decagon, in order to keep it within the square.

100. A regular decagon may also be obtained as follows :

Obtain X, Y, (Fig. 38), as in § 51, dividing AB in median section.

Take M the mid-point of AB.

Fold XC, MO, YD at right angles to AB.

Take 0 in MO such that YO = AY, or YO = XB.

Fig. 37.

Let YO, and XO produced meet XC, and YD in C and D respectively.

Divide the angles XOC and DOY into four equal parts by HOE, KOF, and LOG.

Take OH, OK, OL, OB, OB, and OG equal to OY or OX

Join X, H, K, L, C, D, E, F G, and Y, in order.

As in § 60,

YOX= 2/5 of art.=36°.

Fig. 38.

By bisecting the sides and joining the points thus determined with the center, the perigon is divided into sixteen equal parts. A 16-gon is therefore easily constructed, and so for a 32-gon, and in general a regular 2n-gon.