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 radius across corners of the polygon is that of the distance from O along O B to the horizontal line, giving the length of the side of the polygon, and the width across corners for a given length of one side of the square, is measured by the length of the lines A, B, C, etc. Thus, dotted line 2 shows the length of the side of a nine-sided figure, of 2-inch radius, the radius across corners of the figure being 2 inches.
Fig. 183.
The dotted line 2-1/2 shows the length of the side of a nine-sided polygon, having a radius across corners of 2-1/2 inches. The dotted line 1 shows the diameter, across corners, of a square whose sides measure an inch, and so on.