e line D,
is the measurement across corners, while the length of each side, or
diameter across the flats, is from point O to either of the points 1, or
from either of the points 1 to the dot.

[Illustration: Fig. 182.]

After graphically demonstrating the correctness of the scale we may
simplify it considerably. In Figure 182, therefore, we have applications
shown. A is a hexagon, and if one of its sides be measured, it will be
found that it measures the same as along line 1 from O B to the diagonal
line 45 degrees, which distance is shown by a thickened line.

At 1-1/2 is shown a seven-sided figure, whose diameter is 3 inches, and
radius 1-1/2 inches, and if from the point at 1-1/2 (along the thickened
horizontal line), to the diagonal marked 49 degrees, be measured, it
will be found exactly equal to the length of a side on the polygon.

At C is shown part of a nine-sided polygon, of 2-inch radius, and the
length of one of its sides will be found to equal the distance from the
diagonal line marked 52-1/2 degrees, and the line O B at 2.

Let it now be noted that if from the point O, as a centre, we describe
arcs of circles from the points of division on O B to O P, one end of
each arc will meet the same figure on O P as it started from at O B, as
is shown in Figure 181, and it becomes apparent that in the length of
diagonal line between O and the required arc we have the radius of the
polygon.

EXAMPLE.--What is the radius across corners of a hexagon or six-sided
figure, the length of a side being an inch?

Turning to our scale we find that the place where there is a horizontal
distance of an inch between the diagonal 45 degrees, answering to
six-sided figures, is along line 1 (Figure 182), and the radius of the
circle enclosing the six-sided body is, therefore, an inch, as will be
seen on referring to circle A. But it will be noted that the length of
diagonal line 45 degrees, enclosed between the point O and the arc of
circle from 1 on O B to one on O P, measures also an inch. Hence we may
measure the radius along the diagonal lines if we choose. This, however,
simply serves to demonstrate the correctness of the scale, which, being
understood, we may dispense with most of the lines, arriving at a scale
such as shown in Figure 183, in which the length of the side of the
polygon is the distance from the line O B, measured horizontally to the
diagonal, corresponding to the number of sides of the polygon. The
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