A body has six sides, each side measuring an inch in length, what is its diameter across corners? Find a horizontal line that measures an inch from its intersection of the line o b to the line of 45 degrees, and along this latter to the point o is one-half the diameter across corners.

Example 3.—It is desired to find the diameter across corners of a square whose side is to measure 3 inches. Measure the distance from the 3 on line o p to the 3 on line o b, which will give the required diameter across corners.

This scale lacks, however, one element, in that the diameter across the flats of a regular polygon being given, it will not give the diameter across the corners. This, however, we may obtain by a somewhat similar construction. Thus, in [Fig. 2808], draw the line o b, and divide it into inches and parts of an inch. From these points of division draw horizontal lines; from the point o draw the following lines and at the following angles from the horizontal line o p:—

A line at 75° for polygons having 12 sides.
„72°„10„
„671⁄2°„8„
„60°„6„

From the point o to the numerals denoting the radius of the polygon is the radius across the flats, while from point o to the horizontal line drawn from those numerals is the radius across corners of the polygon.

A hexagon measures 2 inches across the flats, what is its diameter measured across the corners? Now, from point o to the horizontal line marked 1 inch, measured along the line of 60 degrees, is 15⁄32 inches; hence the hexagon measures twice that, or 25⁄16 inches across corners. The proof of the construction is shown in the figure, the hexagon and other polygons being marked for clearness of illustration.

Fig. 2810.