f the
polygon across its corners is given, the circle drawn to that dimension
circumscribes the polygon, because the circle is without or outside of
the polygon and touches it at its corners only. When the dimension
across the flats of the polygon is given, or when the dimension given is
that of a circle that can be inscribed or marked within the polygon,
touching its sides but not passing through them, then the polygon
circumscribes or envelops the circle, and the circle is inscribed or
marked within the polygon. Thus, in Figure 71 _a_, the circle is
inscribed within the polygon, while in Figure 72 the polygon is
circumscribed by the circle; the first is therefore a circumscribed and
the second an inscribed polygon. A regular polygon is one the sides of
which are all of an equal length.

NAMES OF REGULAR POLYGONS.

A figure of 3 sides is called a Trigon.
" 4 " " Tetragon.
polygon 5 " " Pentagon.
" 6 " " Hexagon.
" 7 " " Heptagon.
" 8 " " Octagon.
" 9 " " Enneagon or Nonagon.

[Illustration: Fig. 73.]

[Illustration: Fig. 74.]

The angles of regular polygons are designated by their degrees of angle,
"at the centre" and "at the circumference." By the angle at the centre
is meant the angle of a side to a radial line; thus in Figure 73 is a
hexagon, and at C is a radial line; thus the angle of the side D to C is
60 degrees. Or if at the two ends of a side, as A, two radial lines be
drawn, as B, C, then the angles of these two lines, one to the other,
will be the "angle at the centre." The angle at the circumference is the
angle of one side to its next neighbor; thus the angle at the
circumference in a hexagon is 120 degrees, as shown in the figure for
the sides E, F. It is obvious that as all the sides are of equal length,
they are all at the same angle both to the centre and to one another. In
Figure 74 is a trigon, the angles at its centre being 120, and the angle
at the circumference being 60, as marked.

The angles of regular polygons:

Trigon, at the centre, 120 deg., at the circumference, 60 deg..
Tetragon, " 90 deg., " " 90 deg..
Pentagon, " 72 deg., " " 108 deg..
Hexagon, " 60 deg., " " 120 deg..
Octagon, " 45 deg., " " 135 deg..
Enneagon, "