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_.

[Illustration: 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.

[Illustration: 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