DOI Seite / Zitierlink:
https://doi.org/10.11588/diglit.25888#0043
GEOMETRICAL CONSTRUCTIONS—DRAWING OF ELEMENTARY FORMS.
27
draw perpendiculars, h k, &c., to the diagonals, touching
the circle. The octagon so formed is the figure required.
Or, to find the points h, k, &c., cut the sides from the
corners of the square as described in Problem XXXVI.
Problem XXXIX—To inscribe a circle within a
regular 'polygon.
When the polygon has an even number of sides, as in
(Fig. 115), bisect two opposite sides at A and B, draw A B,
and bisect it at C by a diagonal D E drawn between
opposite angles; with the radius C A describe the circle
as required.
(%. 115.) {Fig. 110.)
D E
When the number of sides is odd, bisect two of the
sides at A and B, and draw lines A E, B D, to the opposite
angles, intersecting at C ; from C with C A as radius,
describe the circle as required.
Problem XL.—To describe a circle without a regular
polygon.
When the number of sides is even, draw two diagonals
from opposite angles, like D E, (Fig. 115), to intersect at C;
and from C with C D as radius, describe the circle
required.
When the number of sides is odd, find the centre C,
(Fig. 116), as in last problem, and with C D as radius
describe the circle.
The foregoing selection of problems on regular figures
are the most useful in mechanical practice, on that subject.
Several other regular figures may be constructed from
them by bisection of the arcs of the circumscribing circles.
In this way a decagon, or ten-sided polygon, may be
formed from the pentagon, as shown by the bisection of
the arc B II at k, in Problem XXXI. Inversely, an
equilateral triangle may be inscribed by joining the alter-
nate points of division found for a hexagon.
The constructions for inscribing regular polygons in
circles, are suitable also for dividing the circumference of
a circle into a number of equal parts. To supply a means
of dividing the circumference into any number of parts,
including cases not provided for in the foregoing problems,
Table oe Polygonal Angles.
Number of Sides of
Regular Polygon; or
number of equal parts
of the circumference.
Angle
at Centre.
Number of Sides of
Regular Polygon.
Angle
at Centre.
No.
Degrees.
No.
Degrees.
3
120
12
30
4
90
13
274
5
72
14
25f
6
60
15
24
7
51|
16
22J
8
45
17
214
9
40
18
20
10
36
19
19
11
--
32x8t
20
18
the annexed table of angles relating to polygons, expressed
in degrees, will be found of general utility. In this table,
the angle at the centre is found by dividing 360°, the
number of degrees in a circle, by the number of sides in
the polygon; and by setting off round the centre of the
circle, a succession of angles by means of the protractor,
equal to the angle in the table due to a given number
of sides, the radii so drawn will divide the circumference
into the same number of parts. The triangles thus formed
are termed the elementary triangles of the polygon.
Problem XLI.—To inscribe any regular polygon in
a given circle; or to divide the circumference into a
given number of equal parts, by means of the angle at
the centre.
Suppose the circle is to contain a hexagon, or is to be
divided at the circumference into six equal parts. Find
the angle at the centre for a liexa-
(%• 117.) 6
gon, or 60°; draw any radius B C,
and set off by a protractor or other-
wise, the angle at the centre C B D
equal to 60°; then the interval C D
is one side of the figure, or segment
of the circumference; and the re-
maining points of division may be
found either by stepping along the
circumference with the distance C D in the dividers, or by
setting off the remaining five angles, of 60° each, round
the centre.
Illustrations of the use of the Drawing Board
AND ITS APPURTENANCES, IN THE CONSTRUCTION OF THE
FOREGOING PROBLEMS.
In the description of the drawing board and T square
it was stated that by applying the stock of the square to
one edge of the board, the blade would lie over the board
at right angles to the edge and parallel to the neighbour-
ing edges, and that by shifting the stock to the neighbour-
ing edge, the blade would assume a position at right
angles to its first position. It thus appears that by slid-
ing the stock along two contiguous edges of the board, the
left hand and bottom edges, any number of parallel and
perpendicular lines may be drawn, within the range of
the board, upon the paper stretched upon it. In so far,
therefore, the T square supersedes the application of all
the problems for drawing parallels and perpendiculars,
coinciding in direction with the edges of the board; for
the square need only be set with its edge coincident with
the points through which the line is to be drawn, and
the pen or pencil drawn along the edge will describe the
line required.
When, indeed, the perpendiculars or upright lines are
of short length, within the range of the set-square, a great
deal of shifting of the T square from the left edge of the
board may be avoided by the use of the triangle, as when
the horizontal edge is applied to that of the blade, short
perpendiculars may be at once described. For this pur-
27
draw perpendiculars, h k, &c., to the diagonals, touching
the circle. The octagon so formed is the figure required.
Or, to find the points h, k, &c., cut the sides from the
corners of the square as described in Problem XXXVI.
Problem XXXIX—To inscribe a circle within a
regular 'polygon.
When the polygon has an even number of sides, as in
(Fig. 115), bisect two opposite sides at A and B, draw A B,
and bisect it at C by a diagonal D E drawn between
opposite angles; with the radius C A describe the circle
as required.
(%. 115.) {Fig. 110.)
D E
When the number of sides is odd, bisect two of the
sides at A and B, and draw lines A E, B D, to the opposite
angles, intersecting at C ; from C with C A as radius,
describe the circle as required.
Problem XL.—To describe a circle without a regular
polygon.
When the number of sides is even, draw two diagonals
from opposite angles, like D E, (Fig. 115), to intersect at C;
and from C with C D as radius, describe the circle
required.
When the number of sides is odd, find the centre C,
(Fig. 116), as in last problem, and with C D as radius
describe the circle.
The foregoing selection of problems on regular figures
are the most useful in mechanical practice, on that subject.
Several other regular figures may be constructed from
them by bisection of the arcs of the circumscribing circles.
In this way a decagon, or ten-sided polygon, may be
formed from the pentagon, as shown by the bisection of
the arc B II at k, in Problem XXXI. Inversely, an
equilateral triangle may be inscribed by joining the alter-
nate points of division found for a hexagon.
The constructions for inscribing regular polygons in
circles, are suitable also for dividing the circumference of
a circle into a number of equal parts. To supply a means
of dividing the circumference into any number of parts,
including cases not provided for in the foregoing problems,
Table oe Polygonal Angles.
Number of Sides of
Regular Polygon; or
number of equal parts
of the circumference.
Angle
at Centre.
Number of Sides of
Regular Polygon.
Angle
at Centre.
No.
Degrees.
No.
Degrees.
3
120
12
30
4
90
13
274
5
72
14
25f
6
60
15
24
7
51|
16
22J
8
45
17
214
9
40
18
20
10
36
19
19
11
--
32x8t
20
18
the annexed table of angles relating to polygons, expressed
in degrees, will be found of general utility. In this table,
the angle at the centre is found by dividing 360°, the
number of degrees in a circle, by the number of sides in
the polygon; and by setting off round the centre of the
circle, a succession of angles by means of the protractor,
equal to the angle in the table due to a given number
of sides, the radii so drawn will divide the circumference
into the same number of parts. The triangles thus formed
are termed the elementary triangles of the polygon.
Problem XLI.—To inscribe any regular polygon in
a given circle; or to divide the circumference into a
given number of equal parts, by means of the angle at
the centre.
Suppose the circle is to contain a hexagon, or is to be
divided at the circumference into six equal parts. Find
the angle at the centre for a liexa-
(%• 117.) 6
gon, or 60°; draw any radius B C,
and set off by a protractor or other-
wise, the angle at the centre C B D
equal to 60°; then the interval C D
is one side of the figure, or segment
of the circumference; and the re-
maining points of division may be
found either by stepping along the
circumference with the distance C D in the dividers, or by
setting off the remaining five angles, of 60° each, round
the centre.
Illustrations of the use of the Drawing Board
AND ITS APPURTENANCES, IN THE CONSTRUCTION OF THE
FOREGOING PROBLEMS.
In the description of the drawing board and T square
it was stated that by applying the stock of the square to
one edge of the board, the blade would lie over the board
at right angles to the edge and parallel to the neighbour-
ing edges, and that by shifting the stock to the neighbour-
ing edge, the blade would assume a position at right
angles to its first position. It thus appears that by slid-
ing the stock along two contiguous edges of the board, the
left hand and bottom edges, any number of parallel and
perpendicular lines may be drawn, within the range of
the board, upon the paper stretched upon it. In so far,
therefore, the T square supersedes the application of all
the problems for drawing parallels and perpendiculars,
coinciding in direction with the edges of the board; for
the square need only be set with its edge coincident with
the points through which the line is to be drawn, and
the pen or pencil drawn along the edge will describe the
line required.
When, indeed, the perpendiculars or upright lines are
of short length, within the range of the set-square, a great
deal of shifting of the T square from the left edge of the
board may be avoided by the use of the triangle, as when
the horizontal edge is applied to that of the blade, short
perpendiculars may be at once described. For this pur-