Let it be required to draw a polygon having twelve equal sides, and the triangle of sixty is used, marking all the lines within the circle in Figure 173, except a, for which the square blade is used; the only lines to be inked in are such as b, c. In this example there is a corner at the top and bottom, but suppose it were required that a flat should fall there instead of a corner; then all we have to do is to set the square blade S at the required angle, as in Figure 174, and then proceed as before, bearing in mind that the point of the circle nearest to the square blade, straight-edge, or whatever the triangle is rested on, is always a corner of a polygon having twelve sides.

Fig. 173.

Fig. 174.

Fig. 175.

In both of these examples we have assumed that the diameter across corners of the polygon was given, but suppose the diameter across the flats were given, and the construction is a little more complicated. Circle a, a, in Figure 175, is drawn of the required diameter across the flats, and the lines of division are drawn across with the triangle of 60 as before; the triangle of 45 is then used to draw the four lines, b, c, d, e, joining the ends of lines i, j, k, l, and touching the inner circle, a, a. The outer circle is then pencilled in, touching the lines of division where they meet the lines b, c, d, e, and the rest of the lines for the sides of the polygon may then be drawn within the outer circle, as at g, h.

Fig. 176.

It is obvious, also, that the triangle may be used to draw slots radiating from a centre, as in Figure 176, where it is desired to draw a chuck-plate having 6 slots. The triangle of 60 is used to draw the centre lines, a, b, c, etc., for the slots. From the centre, the arcs e, f, g, h, etc., are marked, showing where the centres will fall for describing the half circles forming the ends of the slots. Then half circles, i, j, k, l, etc., being drawn, the sides of the slots may be drawn in with the triangle, and the outer circle and the slots inked in.

If the slots are not to radiate from the centre of the circle the process is as follows:

The outer circle a, Figure 177, being drawn, an inner one b is drawn, its radius equalling the amount; the centres of the slots are to point to one side of the centre of circle a. The triangle is then used to divide the circle into the requisite number of divisions c for the slots, and arcs i, j, are then drawn for the lengths of the slots. The centre lines e are then drawn, passing through the lines c, and the arcs i, j, etc., and touching the perimeter of the inner circle b; arcs f, g, are then marked in, and their sides joined with the triangle adjusted by hand. All that would be inked in black are the outer circle and the slots, but the inner circle b and a centre line of one of the slots should be marked in red ink to show how the inclination of the slot was obtained, and therefore its amount.

Fig. 177.

For a five-sided figure it is best to step around the circumference of the circle with the compasses, but for a three-sided one, or trigon, the construction is as follows: It will be 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.

Fig. 178.

Fig. 179.

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.

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:

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.