found that the compasses set to
the radius of a circle will accurately divide it into six equal
divisions, as is shown in Figure 178; hence every other one of these
divisions will be the location for a corner of a trigon.

The circle being drawn, a line A, 179, is drawn through its centre, and
from its intersection with the circle as at _b_, here a step on each
side is marked as _c_, _d_, then lines _c_ to _d_, and _c_ and _d_ to
_e_, where A meets, the circle will describe a trigon. If the figure is
to stand vertical, all that is necessary is to draw the line _a_
vertical, as in Figure 180. A ready method of getting the dimension
across corners, across the flats, or the length of a side of a given
polygon, is by means of diagrams, such as shown in the following
figures, which form excellent examples for practice.

[Illustration: Fig. 178.]

[Illustration: Fig. 179.]

[Illustration: Fig. 180.]

Draw the line O P, Figure 181, and at a right angle to it the line O B;
divide these two lines into parts of one inch, as shown in the cut,
which is divided into inches and quarter inches, and from these points
of division draw lines crossing each other as shown.

[Illustration: Fig. 181.]

From the point O, draw diagonal lines, at suitable angles to the line O
P. As shown in the cut, these diagonal lines are marked:

40 degrees for 5 sided figures.
45 " " 6 " "
49 " " 7 " "
52-1/2 " " 8 " "
55-1/2 " " 9 " "

But still others could be added for figures having a greater number of
sides.

1. Now it will be found as follows: Half the diameter, or the radius of
a piece of cylindrical work being given, and the number of sides it is
to have being stated, the length of one side will be the distance
measured horizontally from the line O B to the diagonal line for that
particular number of sides.

EXAMPLE.--A piece of work is 2-1/2 inches in diameter, and is required
to have 9 sides: what will be the length of the sides or flats?

Now the half diameter or radius of 2-1/2 inches is 1-1/4 inches. Then
look along the line O B for 1-1/4, which is denoted in the cut by
figures and the arrow A; set one point of the compasses at A, and the
other at the point of crossing of the diagonal line with the 1-1/4
horizontal line, as shown in the figure at _a_, and from A to _a_ is the
length of one side.

Again: A piece of work, 4 inches in diameter, is to have 9 sides: how