e the hexagonal figure _K_ we have made use of the vanishing
scale _O_ and the vanishing point _V_. Another method could be used by
drawing two hexagons one over the other at the required height.

CXIX

APPLICATION OF THE HEXAGON TO DRAWING A KIOSK

[Illustration: Fig. 215.]

This figure is built up from the hexagon standing on a rectangular base,
from which we have raised verticals, &c. Note how the jutting portions
of the roof are drawn from _o'_. But the figure explains itself, so
there is no necessity to repeat descriptions already given in the
foregoing problems.

CXX

THE PENTAGON

[Illustration: Fig. 216.]

The pentagon is a figure with five equal sides, and if inscribed in a
circle will touch its circumference at five equidistant points. With any
convenient radius describe circle. From half this radius, marked 1, draw
a line to apex, marked 2. Again, with 1 as centre and 1 2 as radius,
describe arc 2 3. Now with 2 as centre and 2 3 as radius describe arc
3 4, which will cut the circumference at point 4. Then line 2 4 will be
one of the sides of the pentagon, which we can measure round the circle
and so produce the required figure.

To put this pentagon into parallel perspective inscribe the circle in
which it is drawn in a square, and from its five angles 4, 2, 4, &c.,
drop perpendiculars to base and number them as in the figure. Then draw
the perspective square (Fig. 217) and transfer these measurements to its
base. From these draw lines to point of sight, then by their aid and the
two diagonals proceed to construct the pentagon in the same way that we
did the triangles and other figures. Should it be required to place this
pentagon in the opposite position, then we can transfer our measurements
to the far side of the square, as in Fig. 218.

[Illustration: Fig. 217.]

[Illustration: Fig. 218.]

Or if we wish to put it into angular perspective we adopt the same
method as with the hexagon, as shown at Fig. 219.

[Illustration: Fig. 219.]

Another way of drawing a pentagon (Fig. 220) is to draw an isosceles
triangle with an angle of 36 deg at its apex, and from centre of each
side of the triangle draw perpendiculars to meet at _o_, which will be
the centre of the circle in which it is inscribed. From this centre and
with radius _OA_ describe circle A 3 2, &c. Take base of triangle 1 2,
measure it round the circle, and so find the five points through which
to draw the p