Draw the diameter A B, and make the radius D C perpendicular to A B. Bisect D B in E. From E as a centre, and with E C as radius, describe an arc cutting A D in F. Join C F, which will be the side of the pentagon, C D that of the hexagon, and D F that of the decagon. [Fig. 12.]
To find the angles at the centre, and circumference of a regular polygon.
Divide 360 by the number of the sides of the given polygon, and the quotient will be the angle at the centre; and this angle being subtracted from 180, the difference will be the angle, at the circumference, required.
Table, showing the angles at the centre, and circumference.
| Names. | No. of | Angles | Angles at |
| sides. | at centre. | circumference. | |
| Trigon | 3 | 120° | 60° |
| Tetragon | 4 | 90° | 90° |
| Pentagon | 5 | 72° | 108° |
| Hexagon | 6 | 60° | 120° |
| Heptagon | 7 | 51° 25 5′ 7 | 128° 34 2′ 7 |
| Octagon | 8 | 45° | 135° |
| Nonagon | 9 | 40° | 140° |
| Decagon | 10 | 36° | 144° |
To inscribe any regular polygon in a circle.
From the centre C draw the radii C A, C B, making an angle equal to that at the centre of the proposed polygon, as contained in the preceding table. Then the distance A B will be one side of the polygon, which, being carried round the circumference the proper number of times, will complete the polygon required. [Fig. 13.]
Fig. 15-20.